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This paper proposes a family of line--search methods to deal with weighted orthogonal procrustes problems. In particular, the proposed family uses a search direction based on a convex combination between the Euclidean gradient and the Riemannian gradient of the cost function. The non--monotone line--search of Zhang and Hager, and an adaptive Barzilai--Borwein step--size are the chosen tools, in order to speed up the convergence of the new family of methods. One of the extremes of that convex combination is reduced to well--known spectral projected gradient method, while the another one can be interpreted as a Riemannian steepest descent method. To preserve feasibility of all the iterates, we use a projection operator based on the singular value decomposition, which can be computed efficiently via the spectral decomposition of an appropriate matrix. In addition, we prove that the entire uncountable collection of search directions satisfies a descent condition. Some numerical experiments are provided in order to demonstrate the effectiveness of the new approach.

Optimizing warehouse layouts is crucial due to its significant impact on efficiency and productivity. We present an AI-driven framework for automated warehouse layout generation. This framework employs constrained beam search to derive optimal layouts within given spatial parameters, adhering to all functional requirements. The feasibility of the generated layouts is verified based on criteria such as item accessibility, required minimum clearances, and aisle connectivity. A scoring function is then used to evaluate the feasible layouts considering the number of storage locations, access points, and accessibility costs. We demonstrate our method's ability to produce feasible, optimal layouts for a variety of warehouse dimensions and shapes, diverse door placements, and interconnections. This approach, currently being prepared for deployment, will enable human designers to rapidly explore and confirm options, facilitating the selection of the most appropriate layout for their use-case.

Accurately estimating load is essential for effective electric distribution planning, assets management, precise power flow predictions, accurate power losses calculations, and efficient integration of distributed energy resources. This work describes a dataset that was generated using Matlab and OpenDSS to produce several simulations in which load estimation is performed using a direct search method called pattern search. These simulations were conducted on three typical distribution feeders (IEEE 13-bus, 37-bus, and 123-bus) that support studies in distribution planning, assets management, power flow predictions, power losses calculations, and distributed resource integration. The dataset includes individual demand profiles of residential, commercial, and industrial consumers specified for the three distribution feeders, comprising 96 distinct scenarios. An optimization method was developed to obtain the dataset, which employs the pattern search technique to estimate loads through the optimization of objective functions and specified constraints. The load estimation quality was assessed for all three feeders, utilizing estimation quality indices proposed by the authors. These indices evaluated both the initial and proposed load estimation methods across the developed scenarios. Furthermore, the data provided in this article can be utilized for comparison with future load estimation studies, particularly regarding the quality of the method's results.

Abstract The seagull optimization algorithm (SOA) is a meta-heuristic algorithm proposed in 2019. It has the advantages of structural simplicity, few parameters and easy implementation. However, it also has some defects including the three main drawbacks of slow convergence speed, simple search method and poor ability of balancing global exploration and local exploitation. Besides, most of the improved SOA algorithms in the literature have not considered the drawbacks of the SOA comprehensively enough. This paper proposes a hybrid strategies based algorithm (ISOA) to overcome the three main drawbacks of the SOA. Firstly, a hyperbolic tangent function is used to adjust the spiral radius. The spiral radius can change dynamically with the iteration of the algorithm, so that the algorithm can converge quickly. Secondly, an adaptive weight factor improves the position updating method by adjusting the proportion of the best individual to balance the global and local search abilities. Finally, to overcome the single search mode, an improved chaotic local search strategy is introduced for secondary search. A comprehensive comparison between the ISOA and other related algorithms is presented, considering twelve test functions and four engineering design problems. The comparison results indicate that the ISOA has an outstanding performance and a significant advantage in solving engineering problems, especially with an average improvement of 14.67% in solving welded beam design problem.

Teaching–learning based optimization algorithm (TLBO) is an efficient population-based nature-inspired meta-heuristic designed initially to solve unconstrained optimization problems. TLBO being a swam intelligence based algorithm is renowned for fast convergence. This characteristic, however, occasionally causes it to converge too soon to a non-globally optimal solution. Group discussion techniques are used in educational institutions all over the world to help participants to exchange ideas and information and to improve their skills. This work imitates the outlined strategy, therefore innovating TLBO. Here, a group of virtual students is selected corresponding to each student in the learner phase of TLBO to produce diversity in the population and to balance the exploration and exploitation capabilities of it, hence reducing the likelihood of early convergence. GTLBO stands for the new algorithm that incorporates group discussion. It is essential to incorporate some constraint handling techniques (CHTs) into the TLBO framework in order to handle constrained optimization problems (COPs). The most recent, effective, and enhanced version of the widely used CHT stochastic ranking (SR) is called hybrid stochastic ranking (HSR). HSR is added to the GTLBO structure as a selection operator for managing COPs; the resulting setup is referred to as HSR-GTLBO. Additionally, the suggested algorithm’s group size for discussion is sensitive, resulting in a number of constrained variants. The efficacy of these variations is assessed with the suit CEC 2017’s constrained problems. In conformity with the parameters settlement, comparison, and ranking criteria of CEC 2017, the simulation results of the proposed versions are compared and ranked using the most advanced state-of-the-art algorithms. As a result, in the competition, the newly developed HSR-GTLBO variants placed first, second, and third.

This paper proposes the Shrike Optimization Algorithm (SHOA) as a swarm intelligence optimization algorithm. Many creatures, who live in groups and survive for the next generation, randomly search for food; they follow the best one in the swarm, a phenomenon known as swarm intelligence. While swarm-based algorithms mimic the behaviors of creatures, they struggle to find optimal solutions in multi-modal problem competitions. The swarming behaviors of shrike birds in nature serve as the main inspiration for the proposed algorithm. The shrike birds migrate from their territory in order to survive. However, the SHOA replicates the survival strategies of shrike birds to facilitate their living, adaptation, and breeding. Two parts of optimization exploration and exploitation are designed by modeling shrike breeding and searching for foods to feed nestlings until they get ready to fly and live independently. This paper is a mathematical model for the SHOA to perform optimization. The SHOA benchmarked 19 well-known mathematical test functions, 10 from CEC-2019 and 12 from CEC-2022’s most recent test functions, for a total of 41 competitive mathematical test functions and four real-world engineering problems with different conditions, both constrained and unconstrained. The statistical results obtained from the Wilcoxon ranking sum and Fridman test show that SHOA has a significant statistical superiority in handling the test benchmarks compared to competitor algorithms in multi-modal problems. The results for engineering optimization problems show the SHOA outperforms other nature-inspired algorithms in many cases.

Hyperspectral (HS) unmixing is the process of decomposing an HS image into material-specific spectra (endmembers) and their spatial distributions (abundance maps). Existing unmixing methods have two limitations with respect to noise robustness. First, if the input HS image is highly noisy, even if the balance between sparse and piecewise-smooth regularizations for abundance maps is carefully adjusted, noise may remain in the estimated abundance maps or undesirable artifacts may appear. Second, existing methods do not explicitly account for the effects of stripe noise, which is common in HS measurements, in their formulations, resulting in significant degradation of unmixing performance when such noise is present in the input HS image. To overcome these limitations, we propose a new robust HS unmixing method based on constrained convex optimization. Our method employs, in addition to the two regularizations for the abundance maps, regularizations for the HS image reconstructed by mixing the estimated abundance maps and endmembers. This strategy makes the unmixing process much more robust in highly noisy scenarios, under the assumption that the abundance maps used to reconstruct the HS image with desirable spatio-spectral structure are also expected to have desirable properties. Furthermore, our method is designed to accommodate a wider variety of noise including stripe noise. To solve the formulated optimization problem, we develop an efficient algorithm based on a preconditioned primal-dual splitting method, which can automatically determine appropriate stepsizes based on the problem structure. Experiments on synthetic and real HS images demonstrate the advantages of our method over existing methods.

An Error-Bound Particle Swarm Optimization (EB-PSO) is proposed in this work. The objective function is evaluated for each particle in each iteration. The velocity update equation is modified by introducing two new parameters $\zeta _{1}$ and $\zeta _{2}$ . These parameters varies exponentially, within the bounds ( $\zeta _{1,min}$ , $\zeta _{2,min}$ ) and ( $\zeta _{1,max}$ , $\zeta _{2,max}$ ), with respect to the number of iterations. Initially, a higher value of $\zeta _{2}$ and minimum value of $\zeta _{1}$ is chosen to facilitate a global search. Once the global error ( $\varepsilon _{2}$ ) is less than the desired value, $\zeta _{1}$ is allowed to increase from its minimum value and $\zeta _{2}$ is held constant at $\zeta _{2,max}$ . This leads to local exploitation of the search space. The proposed algorithm is implemented on Python platform. To verify the effectiveness of the proposed EB-PSO algorithm in analog circuit sizing, a case study on the performance and optimization of two-stage op-amp is presented, whose validation is done in Cadence-Virtuoso environment at 45-nm CMOS technology. The results show that the proposed EB-PSO algorithm converges in 11 iterations for two-stage op-amp, whereas it takes 23, 29, and 41 iterations to converge for conventional GA, DE, and PSO algorithms respectively.

Drug molecule design is a classic research topic. Drug experts traditionally design molecules relying on their experience. Manual drug design is time-consuming and may produce low-efficacy and off-target molecules. With the popularity of deep learning, drug experts are beginning to use generative models to design drug molecules. A well-trained generative model can learn the distribution of training samples and infinitely generate drug-like molecules similar to the training samples. The automatic process improves design efficiency. However, most existing methods focus on proposing and optimizing generative models. How to discover ideal molecules from massive candidates is still an unresolved challenge. We propose a visualization system to discover ideal drug molecules generated by generative models. In this paper, we investigated the requirements and issues of drug design experts when using generative models, i.e., generating molecular structures with specific constraints and finding other molecular structures similar to potential drug molecular structures. We formalized the first problem as an optimization problem and proposed using a genetic algorithm to solve it. For the second problem, we proposed using a neighborhood sampling algorithm based on the continuity of the latent space to find solutions. We integrated the proposed algorithms into a visualization tool, and a case study for discovering potential drug molecules to make KOR agonists and experiments demonstrated the utility of our approach.

The dislocation hyperbolic augmented Lagrangian algorithm (DHALA) is a new approach to the hyperbolic augmented Lagrangian algorithm (HALA). DHALA is designed to solve convex nonlinear programming problems. We guarantee that the sequence generated by DHALA converges towards a Karush-Kuhn-Tucker point. We are going to observe that DHALA has a slight computational advantage in solving the problems over HALA. Finally, we will computationally illustrate our theoretical results.

Ideal cascades for binary mixtures of isotopes are specified by no-mixing at confluent points and minimum total flows. Studies show that there is another type of cascade called an optimum cascade. These cascades have total flows lower than ideal cascades while separation factors are greater than unity and mixings are allowed. In this paper, using a Co-evolutionary Particle Swarm Optimization (CPSO) algorithm, the ideal and optimum cascades are compared in different operating regimes. The CPSO is a metaheuristic algorithm that uses the concept of co-evolution to deal with constrained engineering optimization problems. In this study, it is used to find the parameters of the optimum cascade. In this work, three test cases are considered to compare ideal and optimum cascades. The first test case includes two distinct symmetrical separation cascades. In the first cascade, the minimum total flow for the ideal type 3 cascade and its corresponding optimum cascade is obtained as ∑L⁄P=176.7128, and in the second one for the ideal type 1 cascade and its corresponding optimum cascade, it is obtained as ∑L⁄P=202.7828. The results show that for symmetric separation, the ideal cascade coincides with the optimum cascade. In test case 2, the minimum total flows for the ideal type 1 cascade of non-symmetrical separation elements and its corresponding optimum cascade (CPSO) are obtained as ∑L⁄P=477.6170 and ∑L⁄P=228.6997, respectively. In test case 3, for the ideal type 2 cascade of non-symmetrical elements and its corresponding optimum cascade, the minimum total flows are obtained as ∑L⁄P=299.99 and ∑L⁄P=191.6584, respectively.

Introduction. Nonsmooth optimization problems arise in a wide range of applications, including engineering, finance, and deep learning, where activation functions often have discontinuous derivatives, such as ReLU. Conventional optimization algorithms developed primarily for smooth problems face difficulties when applied to nonsmooth contexts due to discontinuities and other associated irregularities. Possible approaches to overcome these problems include smoothing of functions and applying non-smooth optimization techniques. In particular, Shor's r-algorithm (Shor, Zhurbenko (1971), Shor (1979)) with space stretching in the direction of the difference of two subsequent subgradients is a competitive non-smooth optimization method (Bagirov et al. (2014)). However, the original r-algorithm is designed to minimize unconstrained convex functions. The goal of the work. The standard technique for applying this algorithm to problems with constraints consists in the use of exact non-smooth penalty functions (Eremin (1967), Zangwill (1967)). At the same time, it is necessary to correctly choose (quite large) the penalty coefficient of the penalty functions. Norkin (2020, 2022), Galvan et al. (2021) propose the so-called projective exact penalty functions method, which theoretically does not require choice of the penalty coefficient. The purpose of the present work is to study an applicability of the new exact projective non-smooth penalty functions method for solving conditional problems of non-smooth optimization by Shor's r-algorithm. The results. In this paper, the original optimization problem with convex constraints is first transformed into an unconstrained problem by the projective penalty function method, and then the r-algorithm is used to solve the transformed problem. The results of testing this approach on problems with linear constraints using a program implemented in Matlab are presented. The results of the present study show that the standard method of non-smooth penalties combined with Shor's r-algorithm is fast, due to the use of the provided program to calculate the subgradients, but it requires the correct selection of the penalty parameter. The projective penalty method is slow because in this study it uses finite differences to calculate the gradients, but it is quite stable with respect to the choice of the penalty parameter. Further research will be aimed at investigating the differential properties of the projection mapping and reducing the time of computing subgradients for account of parallel calculations.

Feature detection and matching is the key technique for remote sensing image processing and related applications. In this paper, a PDE-constrained optimization model is proposed to determine the scale levels advantageous for feature detection. A variance estimation technique is introduced to treat the observation optical images polluted by additive zero-mean Gaussian noise and determine the parameter of a nonlinear scale space governed by the partial differential equation. Additive Operator Splitting is applied to efficiently solve the PDE constraint, and an iterative algorithm is proposed to approximate the optimal subset of the original scale level set. The selected levels are distributed more uniformly in the total variation sense and helpful for generating more accurate and robust feature points. The experimental results show that the proposed method can achieve about a 30% improvement in the number of correct matches with only a small increase in time cost.

Due to the pervasive generalization challenges in optimization technology, there is a noticeable trend toward planning and diversifying optimization techniques. This paper focuses on particle swarm optimization algorithms, particularly their application in multi-objective optimization scenarios. Initially, the study examines basic particle swarm, standard particle swarm, and particle swarm algorithms with a shrinkage factor. Subsequently, an enhanced particle swarm optimization algorithm is proposed, incorporating a hybridization model and a convergence factor model tailored to the specific characteristics of particle swarm algorithms. This improved algorithm is then applied to multi-objective optimization problems, establishing a novel algorithm based on the fusion of the enhanced particle swarm approach with constrained optimization. Simulation experiments conducted on this model reveal significant findings. In low-dimensional settings, the algorithm achieves a 100% optimization success rate, marking an average improvement of 53.80%, 40.78%, and 24.76% over competing algorithms. Moreover, in multi-objective optimization simulation experiments, this algorithm generates 142 and 135 optimal solutions, outperforming traditional algorithms by 112 and 107 solutions, respectively. These results validate the efficiency and enhanced performance of the improved particle swarm-based multi-objective optimization algorithm, demonstrating its potential as an effective tool for addressing real-world optimization challenges.

Context: Nowadays, inventory management poses a challenge given the constant demands related to temporality, geographic location, price variability, and budget availability, among others. In neighborhood shops, this process is manually done based on experience (the data generated are ignored), which is sometimes not enough to respond to changes. This shows the need to develop new strategies and tools that use data analysis techniques. Method: Our methodology predicts the weekly demand for 14 common products in neighborhood stores, which is later refined based on investment capital. The method is validated using a database built with synthetic information extracted from statistical sampling. For the prediction model, three supervised learning models are used: support vector machines (SVM), AutoRegressive models (Arx), and Gaussian processes (GP). This work proposes a restricted linear model given an inversion and the predicted quantity of products; the aim is to refine the prediction while maximizing the shopkeeper's profit. Finally, the problem is solved by applying an integer linear programming paradigm. Results: Tests regarding the prediction and inventory adjustment stages are conducted, showing that the methodology can predict the temporal dynamics of the data by inferring the statistical moments of the distributions used. It is shown that it is possible to obtain a maximum profit with a lower investment. Conclusions: Our method allows predicting and refining inventory management in a neighborhood store model where quantities are managed to maximize the shopkeeper's profits. This opens the way to explore this approach in a real scenario or to introduce new techniques that can improve its performance.

This paper constructs a model of the particle swarm algorithm, compares and analyzes the performance of the particle swarm algorithm under the two parameters of w and k in detail, and solves the constrained optimization problem by the particle swarm algorithm. On the basis of the local optimal value to find the global optimal value, the particle swarm algorithm is improved with reference to the particle’s motion state and behavior. Based on the particle swarm algorithm, a digital enterprise management system is constructed to plan enterprise management operations and optimize efficiency. Finally, we compare the performance of different algorithms in enterprise management risk prediction, analyze the correlation between the management system and enterprise management efficiency, and compare the management efficiency of different enterprises to explore the effect of the particle swarm algorithm in digital enterprise management. The results show that the predictive classification effect of the particle swarm algorithm model reaches more than 95% correct rate, and the management system of the particle swarm algorithm presents significance at 1% and 5% significance level for enterprise management efficiency, respectively.

AbstractThe opioid epidemic is an ongoing public health crisis. In North Carolina, overdose deaths due to illicit opioid overdose have sharply increased over the last 5–7 years. Buprenorphine is a U.S. Food and Drug Administration approved medication for treatment of opioid use disorder and is obtained by prescription. Prior to January 2023, providers had to obtain a waiver and were limited in the number of patients that they could prescribe buprenorphine. Thus, identifying counties where increasing buprenorphine would yield the greatest overall reduction in overdose death can help policymakers target certain geographical regions to inform an effective public health response. We propose a Bayesian spatio-temporal model that relates yearly, county-level changes in illicit opioid overdose death rates to changes in buprenorphine prescriptions. We use our model to forecast the statewide count and rate of illicit opioid overdose deaths in future years, and we use nonlinear constrained optimization to identify the optimal buprenorphine increase in each county under a set of constraints on available resources. Our model estimates a negative relationship between death rate and increasing buprenorphine after accounting for other covariates, and our identified optimal single-year allocation strategy is estimated to reduce opioid overdose deaths by over 5%. Supplementary materials for this article are available online.

Handling a large number of geometric constraints brings a big challenge to the surrogate-based aerodynamic shape optimization (ASO) driven by computational fluid dynamics (CFD). It is not feasible to calculate the geometric constraint functions directly during the sub-optimization of a surrogate-based optimization, as the geometric constraint functions are to be evaluated thousands of times for each updating cycle and the total cost of a number of cycles can be prohibitive. This article proposes an efficient method of handling geometric constraints within the framework of a surrogate-based optimization to address this problem. The core idea is to use the Kreisselmeier-Steinhauser (KS) method to aggregate all geometric constraints into one that can be approximated by a cheap surrogate model, in order to avoid the large computational cost associated with tremendous calculation of geometric constraint values. The proposed method is verified by an analytical test case. Then, the proposed method is demonstrated and compared with the methods of building surrogate models of all geometric constraints and calculating all geometric constraints directly during each sub-optimization by drag minimizations of NACA0012 air foil and ONERA M6 wing in transonic flows. To investigate the ability of the proposed method for handling various geometric constraints, drag minimization of CRM wing in viscous transonic flow driven by CFD is performed. Results show that the proposed method can dramatically improve the optimization efficiency of ASO with the number of geometric constraints ranging from 15 to 1429 and the number of types of geometric constraints up to 3, which offers great potential for handling a larger number and more types of geometric constraints.

The number of research works on dynamic constrained optimization problems has been increasing rapidly over the past two decades. In this domain, many real-life decision problems need to be solved repeatedly with changing data and parameters. However, no research on dynamic problems with changes in the coefficients of the constraint functions has been reported. In this paper, to deal with such problems, a new evolutionary framework with multiple novel mechanisms is proposed. The new mechanisms are for (1) dealing with both linear and non-linear components in the constraint functions, (2) identifying the rate of change in the coefficients of the variables and (3) updating the population efficiently after every change occurs in the problem. To evaluate the performance of the proposed algorithm, we designed a new set of 13 dynamic benchmark problems, each of which consists of 20 dynamic changes and 3 different scenarios. The results demonstrate that the proposed algorithm significantly contributes in achieving good quality solutions, high feasibility rates and fast convergence in rapidly changing environments. In addition, the framework shows its capability of using different meta-heuristics to solve dynamic problems.

The optimal power flow problem in power systems is characterized by a number of complex objectives and constraints, which aim to optimize the total fuel cost, emissions, active power loss, voltage magnitude deviation, and other metrics simultaneously. These conflicting objectives and strict constraints challenge existing optimizers in balancing between active power and reactive power, along with good trade-offs among many metrics. To address these difficulties, this paper develops a co-evolutionary algorithm to solve the constrained many-objective optimization problem of optimal power flow, which evolves three populations with different selection strategies. These populations are evolved towards different parts of the huge objective space divided by large infeasible regions, and the cooperation between them renders assistance to the search for feasible and Pareto-optimal solutions. According to the experimental results on benchmark problems and the IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems, the proposed algorithm is superior over peer algorithms in solving constrained many-objective optimization problems, especially the optimal power flow problems.

This work introduces a Partition Bound Particle Swarm Optimization (PB-PSO) algorithm to enhance convergence rates in analog circuit optimization. Two new parameters, $\zeta _{1}$ and $\zeta _{2}$ , are incorporated to adaptively update particle velocities based on iteration numbers. The parameter $\zeta _{1}$ depends on the non-linear convergence factor ( $\alpha $ ) and the number of iterations, $N$ . The results indicate that $\zeta _{1}$ ’s optimal value occurs with $\alpha = 4.~\zeta _{2}$ partitions iterations into two regions, aiding local and global search. The PB-PSO algorithm, implemented in Python, demonstrates higher convergence rates than existing methods, with successful designs verified through Cadence-Virtuoso circuit simulations. The proposed PB-PSO algorithm converges in 15 and 13 iterations for differential amplifier and two-stage op-amp respectively. For a case study of two-stage amplifier, it achieves a gain of 60.4 dB with a phase margin of 79.76°, meeting input specifications within constraints. The figure of merit was then evaluated using the obtained parameters, which turns out to be 0.275 $V^{-2}$ .

Traditional approaches for multi-agent navigation consider the environment as a fixed constraint, despite the obvious influence of spatial constraints on agents' performance. Yet hand-designing conducive environments is inefficient and potentially expensive. The goal of this article is to consider the obstacle layout of the environment as a decision variable in a system-level optimization problem. In other words, we aim to find an automated solution that optimizes the obstacle layout to improve the performance of multi-agent navigation, under a variety of realistic constraints. Towards this end, we propose novel problems of unprioritized and prioritized environment optimization, where the former considers agents unbiasedly and the latter incorporates agent priorities into optimization. We show, through formal proofs, under which conditions the environment can change to guarantee completeness (i.e., all agents reach goals), and analyze the role of agent priorities in the environment optimization. We proceed to impose constraints on the environment optimization that correspond to real-world restrictions on obstacle changes, and formulate it mathematically as a constrained stochastic optimization problem. Since the relationship between agents, environment and performance is challenging to model, we leverage reinforcement learning to develop a model-free solution and a primal-dual mechanism to handle constraints. Distinct information processing architectures are integrated for various implementation scenarios, including online/offline optimization and discrete/continuous environment. Numerical results corroborate the theory and demonstrate the validity and adaptability of our approach.

This work extends a near feasibility threshold (NFT) based adaptive penalty function for constrained multiobjective optimization. The NFT zone adjoining the feasible region is considered as good one, where infeasible solutions are relatively less penalized. The modified penalty function, denoted by TAP with five different settings of NFT is embedded in a renowned multiobjective evolutionary algorithm based on decomposition, MOEA/D. This offers five constrained variants, namely CMOEA/D-TAP1 to CMOEA/D-TAP5, of the base algorithm. These variants are tested on well-known constrained multiobjective benchmark test suits, the CTP series and the CF series. The proposed variants are compared with the four best performing algorithms through HV (hyper volume) metric statistics for CTP series, and with seven state-of-the-art algorithms through Wilcoxon rank sum test employed to mean values of both HV and IGD (inverted generational distance) metrics for CF series. Simulation results reflect that overall performance of the newly introduced variants is better than the competitors for the taken benchmark test suits.

The Multi-Swarm approach allows the use of multiple configurations between two or more populations of particles, where each one can present different approaches (e.g. lbest, gbest, Unified, Guaranteed-Convergence) directed towards improving the optimization process. This article presents a proposal for local/global stochastic interconnection applied to the context of the Multi-Swarm algorithm, as well as for incrementing a local search method for refining previously obtained solutions. Two proposals are introduced for this new Multi-Swarm PSO (MSO). The first one is the inclusion of “counterpart particles”, which establishes a sub-topology between inter-swarm particles, accessed by migration and evaluability rules. The other involves using customized crossover operators and is based on the BLX scheme (Blend Crossover) with direction information used as a reference for establish a subspace search around the particles. Performance and robustness of the new approaches were assessed by ten constrained engineering design optimization problems (COPs), as is compared to other solutions already published in the scientific literature. Results indicate significant performance improvements for all 10 COPs when compared to concurrent-based MSOs. By making available new references from other swarms, the counterpart particles approach tends to improve the optimization process in the search space, while an intermediate layer of local search based on a modified directed BLX crossover should provide an extra search around the particle, and thus, refining previously obtained solutions.

Abstract Clinical information collected in electronic health records (EHRs) is becoming an essential source to emulate randomized experiments. Since patients do not interact with the healthcare system at random, the longitudinal information in large observational databases must account for irregular visits. Moreover, we need to also account for subject-specific unmeasured confounders which may act as a common cause for treatment assignment mechanism (e.g. glucose-lowering medications) while also influencing the outcome (e.g. Hemoglobin A1c). We used the calibration of longitudinal weights to improve the finite sample properties and to account for subject-specific unmeasured confounders. A Monte Carlo simulation study is conducted to evaluate the performance of calibrated inverse probability estimators using time-dependent treatment assignment and irregular visits with subject-specific unmeasured confounders. The simulation study showed that the longitudinal weights with calibrated restrictions improved the finite sample bias when compared to the stabilized weights. The application of the calibrated weights is demonstrated using the exposure of glucose lowering medications and the longitudinal outcome of Hemoglobin A1c. Our results support the effectiveness of glucose lowering medications in reducing Hemoglobin A1c among type II diabetes patients with elevated glycemic index ( $$\ge 8.5 \%$$ ≥ 8.5 % ) using stabilized and calibrated weights.

Bayesian method is commonly used in the reliability assessment of repairable systems with small samples in fields such as aerospace.The knowledge about the informative prior and/or failure data from repairable systems may not be collected as precisely as possible due to uncertain factors,but can be obtained in terms of lower and upper bounds.For the case of such interval uncertainty information,the Bayesian reliability analysis method of multiple repairable whose failure process can follow the power law process(PLP)is studied in this paper.Under the informative priors,the interval-based Bayesian posterior analysis of the PLP is essentially transformed into solving the constrained optimization problem in which the objective function is just the traditional Bayesian posterior analysis result of the PLP under the informative priors.The feasibility and effectiveness of the proposed method is verified with specific enginee-ring examples.The results show that the proposed interval-based Bayesian method of the PLP can provide a refe-rence method for the reliability assessment of repairable systems with small samples in consideration of the influence of uncertain factors.

The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.

Modern additive technologies make it possible to create structures of variable thickness and of any shape. Thus, designers face problems of optimal design of a new type, and these are problems of topological optimization. Such problems are to determine the optimal form of the structure or the optimal distribution of material over the structure. As a rule, the criterion of optimality is the mass of the structure. However, the structure must retain its bearing capacity under a certain load. The symmetric two-shear adhesive joint of the main plate with two overlays of the same shape on both sides is the object of study in this article. The main goal of this study was to determine the optimal form of overlays with variable thicknesses under certain restrictions. The main restriction is the strength of the structure. Furthermore, additional restrictions are imposed on the minimum and maximum thickness of the overlay. Therefore, the solution to the problem is presented in the form of a set of the following tasks: building a mathematical model of the adhesive joint, building a numerical solution to the primal problem using the finite difference method, and building a genetic optimization algorithm. In the presented article, to improve the convergence of the genetic algorithm is proposed to use an island model that consists of several populations. The main feature of the proposed model of the genetic algorithm lies in the fact that on one of the "islands" mutations occur more frequently and with higher dispersion than on the other two "islands". On the one hand, this decision ensures a high rate of evolutionary selection, and on the other hand, the stability of the results is achieved. Several modeling problems are solved in this article. The main results of this research include the following: nonlinear dependence of the overlay length on the applied load was determined; restrictions on the minimum thickness of the overlay, which cause the appearance of a certain “plateau” at the edge of the overlay, the thickness of which is equal to the minimum allowable were defined.

The paper proposes a novel nature-inspired optimization technique called Eagle Perching Optimizer (EPO). It is an addition to the family of swarm-based meta-heuristic algorithms. It mimics eagles’ perching nature to find prey (food). The EPO is based on the exploration and exploitation of an eagle when it descends from the height such that it formulates its trajectory in a way to get to the optimal solution (prey). The algorithm takes bigger chunks of search space and looks for the optimal solution. The optimal solution in that chunk becomes the search space for the next iteration, and this process is continuous until EPO converges to the optimal global solution. We performed the theoretical analysis of EPO, which shows that it converges to the optimal solution. The simulation includes three sets of problems, i.e., uni-model, multi-model, and constrained real-world problems. We employed EPO on the benchmark problems and compared the results with state-of-the-art meta-heuristic algorithms. For the real-world problems, we used a cantilever beam, three-bar truss, and gear train problems to test the robustness of EPO and later made the comparison. The comparison shows that EPO is comparable with other known meta-heuristic algorithms.

The standard approach to encoding constraints in quantum optimization is the quadratic penalty method. Quadratic penalties introduce additional couplings and energy scales, which can be detrimental to the performance of a quantum optimizer. In quantum annealing experiments performed on a D-Wave Advantage, we explore an alternative penalty method that only involves linear Ising terms and apply it to a customer data science problem. Our findings support our hypothesis that the linear Ising penalty method should improve the performance of quantum optimization compared to using the quadratic penalty method due to its more efficient use of physical resources. Although the linear Ising penalty method is not guaranteed to exactly implement the desired constraint in all cases, it is able to do so for the majority of problem instances we consider. For problems with many constraints, where making all penalties linear is unlikely to be feasible, we investigate strategies for combining linear Ising penalties with quadratic penalties to satisfy constraints for which the linear method is not well-suited. We find that this strategy is most effective when the penalties that contribute most to limiting the dynamic range are removed.

Abstract The constrained optimization problems can be transformed into multi-objective optimization problems, and thus can be optimized by multi-objective evolutionary algorithms. This method has been successfully used to solve the constrained optimization problems. However, little theoretical work has been done on the performance of multi-objective evolutionary algorithms for the constrained optimization problems. In this paper, we theoretically analyze the performance of a multi-objective evolutionary algorithms on the constrained minimum spanning tree problem. First, we theoretically prove that the multi-objective evolutionary algorithm is capable of finding a (2,1)-approximation solution for the constrained minimum spanning tree problem in a pseudopolynomial runtime. Then, this simple multi-objective evolutionary algorithm is shown to be efficient on a constructed instance of the problem.

This paper presents an algorithm and optimization procedure for the optimization of the outer rotor structure of the brushless DC (BLDC) motor. The optimization software was developed in the Delphi Tiburón development environment. The optimization procedure is based on the salp swarm algorithm. The effectiveness of the developed optimization procedurewas compared with genetic algorithm and particle swarmoptimization algorithm. The mathematical model of the device includes the electromagnetic field equations taking into account the non-linearity of the ferromagnetic material, equations of external supply circuits and equations of mechanical motion. The external penalty function was introduced into the optimization algorithm to take into account the non-linear constraint function.

The primary objective of this paper is the custom design of an effective, yet relatively easyto- implement, predictive control algorithm to maintain normoglycemia in patients with type 1 diabetes. The proposed patient-tailorable empirical model featuring the separated feedback dynamics to model the effect of insulin administration and carbohydrate intake was proven to be suitable for the synthesis of a high-performance predictive control algorithm for artificial pancreas.Within the introduced linear model predictive control law, the constraints were applied to the manipulated variable in order to reflect the technical limitations of insulin pumps and the typical nonnegative nature of the insulin administration. Similarly, inequalities constraints for the controlled variable were also assumed while anticipating suppression of hypoglycemia states during the automated insulin treatment. However, the problem of control infeasibility has emerged, especially if one uses too tight constraints of the manipulated and the controlled variable concurrently. To this end, exploiting the Farkas lemma, it was possible to formulate the helper linear programming problem based on the solution of which this infeasibility could be identified and the optimality of the control could be restored by adapting the constraints. This adaptation of constraints is asymmetrical, thus one can force to fully avoid hypoglycemia at the expense of mild hyperglycemia. Finally, a series of comprehensive in-silico experiments were carried out to validate the presented control algorithm and the proposed improvements. These simulations also addressed the control robustness in terms of the intersubject variability and the meal announcements uncertainty.

Introduction: It is well known that the common chimpanzee, as both the closest living relative to humans and a facultative bipedal, has the capability of bipedal standing but cannot do so fully upright. Accordingly, they have been of exceeding significance in elucidating the evolution of human bipedalism. There are many reasons why the common chimpanzee can only stand with its hips–knees bent, such as the distally oriented long ischial tubercle and the almost absent lumbar lordosis. However, it is unknown how the relative positions of their shoulder–hip–knee–ankle joints are coordinated. Similarly, the distribution of the biomechanical characteristics of the lower-limb muscles and the factors that affect the erectness of standing as well as the muscle fatigue of the lower limbs remain a mystery. The answers are bound to light up the evolutional mechanism of hominin bipedality, but these conundrums have not been shed much light upon, because few studies have comprehensively explored the effects of skeletal architecture and muscle properties on bipedal standing in common chimpanzees.Methods: Thus, we first built a musculoskeletal model comprising the head-arms-trunk (HAT), thighs, shanks, and feet segments of the common chimpanzee, and then, the mechanical relationships of the Hill-type muscle-tendon units (MTUs) in bipedal standing were deduced. Thereafter, the equilibrium constraints were established, and a constrained optimization problem was formulated where the optimization objective was defined. Finally, thousands of simulations of bipedal standing experiments were performed to determine the optimal posture and its corresponding MTU parameters including muscle lengths, muscle activation, and muscle forces. Moreover, to quantify the relationship between each pair of the parameters from all the experimental simulation outcomes, the Pearson correlation analysis was employed.Results: Our results demonstrate that in the pursuit of the optimal bipedal standing posture, the common chimpanzee cannot simultaneously achieve maximum erectness and minimum muscle fatigue of the lower limbs. For uni-articular MTUs, the relationship between muscle activation, relative muscle lengths, together with relative muscle forces, and the corresponding joint angle is generally negatively correlated for extensors and positively correlated for flexors. For bi-articular MTUs, the relationship between muscle activation, coupled with relative muscle forces, and the corresponding joint angles does not show the same pattern as in the uni-articular MTUs.Discussion: The results of this study bridge the gap between skeletal architecture, along with muscle properties, and biomechanical performance of the common chimpanzee during bipedal standing, which enhances existing biomechanical theories and advances the comprehension of bipedal evolution in humans.

There are many problems in physics, biology, and other natural sciences in which symbolic regression can provide valuable insights and discover new laws of nature. Widespread deep neural networks do not provide interpretable solutions. Meanwhile, symbolic expressions give us a clear relation between observations and the target variable. However, at the moment, there is no dominant solution for the symbolic regression task, and we aim to reduce this gap with our algorithm. In this work, we propose a novel deep learning framework for symbolic expression generation via variational autoencoder (VAE). We suggest using a VAE to generate mathematical expressions, and our training strategy forces generated formulas to fit a given dataset. Our framework allows encoding apriori knowledge of the formulas into fast-check predicates that speed up the optimization process. We compare our method to modern symbolic regression benchmarks and show that our method outperforms the competitors under noisy conditions. The recovery rate of SEGVAE is 65% on the Ngyuen dataset with a noise level of 10%, which is better than the previously reported SOTA by 20%. We demonstrate that this value depends on the dataset and can be even higher.

Summary: Background: The worldwide control rate for hypertension is dismal. An inadequate number of physicians to treat patients with hypertension is one key obstacle. Innovative health system approaches such as delegation of basic tasks to non-physician health workers (task-sharing) might alleviate this problem. Massive scale up of population-wide hypertension management is especially important for low- and middle-income countries such as India. Methods: Using constrained optimization models, we estimated the hypertension treatment capacity and salary costs of staff involved in hypertension care within the public health system of India and simulated the potential effects of (1) an increased workforce, (2) greater task-sharing among health workers, and (3) extended average prescription periods that reduce treatment visit frequency (e.g., quarterly instead of monthly). Findings: Currently, only an estimated 8% (95% uncertainty interval 7%–10%) of ∼245 million adults with hypertension can be treated by physician-led services in the Indian public health system (assuming the current number of health workers, no greater task-sharing, and monthly visits for prescriptions). Without task-sharing and with continued monthly visits for prescriptions, the least costly workforce expansion to treat 70% of adults with hypertension would require ∼1.6 (1.0–2.5) million additional staff (all non-physicians), with ∼INR 200 billion (≈USD 2.7 billion) in additional annual salary costs. Implementing task-sharing among health workers (without increasing the overall time on hypertension care) or allowing a 3-month prescription period was estimated to allow the current workforce to treat ∼25% of patients. Joint implementation of task-sharing and a longer prescription period could treat ∼70% of patients with hypertension in India. Interpretation: The combination of greater task-sharing and extended prescription periods could substantially increase the hypertension treatment capacity in India without any expansion of the current workforce in the public health system. By contrast, workforce expansion alone would require considerable, additional human and financial resources. Funding: Resolve to Save Lives, an initiative of Vital Strategies, was funded by grants from Bloomberg Philanthropies; the Bill and Melinda Gates Foundation; and Gates Philanthropy Partners (funded with support from the Chan Zuckerberg Foundation).

The closed-loop supply chain (CLSC) plays an important role in sustainable development and can help to increase the economic benefits of enterprises. The optimization for the CLSC network is a complicated problem, since it often has a large problem scale and involves multiple constraints. This paper proposes a general CLSC model to maximize the profits of enterprises by determining the transportation route and delivery volume. Due to the complexity of the multi-constrained and large-scale model, a genetic algorithm with two-step rank-based encoding (GA-TRE) is developed to solve the problem. Firstly, a two-step rank-based encoding is designed to handle the constraints and increase the algorithm efficiency, and the encoding scheme is also used to improve the genetic operators, including crossover and mutation. The first step of encoding is to plan the routes and predict their feasibility according to relevant constraints, and the second step is to set the delivery volume based on the feasible routes using a rank-based method to achieve greedy solutions. Besides, a new mutation operator and an adaptive population disturbance mechanism are designed to increase the diversity of the population. To validate the efficiency of the proposed algorithm, six heuristic algorithms are compared with GA-TRE by using different instances with three problem scales. The results show that GA-TRE can obtain better solutions than the competitors, especially on large-scale instances.

With day by day increase in electrical energy demand and uneven distribution of sources in nature, there is a need of Integration of power plants. This integration needs a proper scheduling of all connected generating units in accordance with the variation in load demand. An optimum sharing of load is necessary to minimize the generation cost. Emission is also an important issue in a generation. An attempt is also made in this paper to analyze emission dispatch with the Economic Dispatch optimization. Environmental effects tend to involve renewable sources based power generation. Wind and Solar are most popular and highly abundant among all renewable sources. But, the fluctuations of these sources complicate the load dispatch optimization. Also, the conventional thermal generators itself affected by certain constraints and non linearity such as valve-point loading effect. A proper planning should involve consideration of all this issue, which requires advance soft computing technique. Previous approaches need proper tuning of their specific parameters, to remove this ambiguity, a new Jaya Algorithm technique is introduced in this paper.

In this work, we propose a variation of the cellular particle swarm optimization algorithm with differential evolution hybridization (CPSO-DE) to include constrained optimization, named Ts-CPD. It is implemented as a kernel of electronic design automation (EDA) tool capable of sizing circuit components considering a single-objective design with restrictions and constraints. The aim is to improve the optimization solutions in the sizing of analog circuits. To evaluate our proposal’s performance, we present the design of three analog circuits: a differential amplifier, a two-stage operational amplifier (op-amp), and a folded cascode operational transconductance amplifier. Numerical simulation results indicate that Ts-CPD can find better solutions, in terms of the design objective and the accomplishment of constraints, than those reported in previous works. The Ts-CPD implementation was performed in Matlab using Ngspice and can be found on GitHub (see Data Availability Statement).

Portfolio optimization is a mathematical formulation whose objective is to maximize returns while minimizing risks. A great deal of improvement in portfolio optimization models has been made, including the addition of practical constraints. As the number of shares traded grows, the problem becomes dimensionally very large. In this paper, we propose the usage of modified biogeography-based optimization to solve the large-scale constrained portfolio optimization. The results indicate the effectiveness of the method used.

Dynamic multi-objective optimization problems (DMOPs) are optimization problems where elements of the problems, such as the objective functions and/or constraints, change with time. These problems are characterized by two or more objective functions, where at least two objective functions are in conflict with one another. When solving real-world problems, the incorporation of human decision-makers (DMs)’ preferences or expert knowledge into the optimization process and thereby restricting the search to a specific region of the Pareto-optimal Front (POF) may result in more preferred or suitable solutions. This study proposes approaches that enable DMs to influence the search process with their preferences by reformulating the optimization problems as constrained problems. The subsequent constrained problems are solved using various constraint handling approaches, such as the penalization of infeasible solutions and the restriction of the search to the feasible region of the search space. The proposed constraint handling approaches are compared by incorporating the approaches into a differential evolution (DE) algorithm and measuring the algorithm’s performance using both standard performance measures for dynamic multi-objective optimization (DMOO), as well as newly proposed measures for constrained DMOPs. The new measures indicate how well an algorithm was able to find solutions in the objective space that best reflect the DM’s preferences and the Pareto-optimality goal of dynamic multi-objective optimization algorithms (DMOAs). The results indicate that the constraint handling approaches are effective in finding Pareto-optimal solutions that satisfy the preference constraints of a DM.

A well-known first-order necessary condition for a point to be a local minimizer of a given function is the non-negativity of the dot product of the gradient and a vector in a feasible direction. This paper proposes a series of alternative first-order necessary conditions and corresponding first-order sufficient conditions that seem not to appear in standard texts. The conditions assume a nonzero gradient. The methods use extensions of the notions of gradient, differentiability, and twice differentiability. Examples, including one involving the Karush–Kuhn–Tucker (KKT) theorem, illustrate the scope of the conditions.

The quantum-inspired genetic algorithm (QGA), which combines quantum mechanics concepts and GA to enhance search capability, has been popular and provides an efficient search mechanism. This paper proposes a modified QGA, called dynamic QGA (DQGA). The proposed algorithm utilizes a lengthening chromosome strategy for a balanced and smooth transition between exploration and exploitation phases to avoid local optima and premature convergence. Apart from that, a novel adaptive look-up table for rotation gates is presented to boost the algorithm’s optimization abilities. To evaluate the effectiveness of these ideas, DQGA is tested by various mathematical benchmark functions as well as real-world constrained engineering problems against several well-known and state-of-the-art algorithms. The obtained results indicate the merits of the proposed algorithm and its superiority for solving multimodal benchmark functions and real-world constrained engineering problems.

Hospitals are among the most important institutions of today. For hospitals, efficient use of operating rooms is of great importance. Efficient use of operating rooms is a problem that needs to be solved. The operating room scheduling problem is a very complex problem with large number of constraints. This type of problem called as NP-Hard type problem. NP-Hard type problems do not consist of polynomial values. Therefore, the solution of these problems is very complex and difficult. Solutions consisting of polynomial values can be solved effectively with existing mathematical methods. However, more effective algorithms were needed to solve NP-hard type problems. As a result of the studies, many heuristic, meta-heuristic algorithms such as Genetic Algorithm, Particle Swarm Optimization, Simulated Annealing, Taboo Search Algorithm have been developed to solve the complexity of NP-Hard problems. In this article, the operating room scheduling problem solved with a hybrid genetic algorithm. In this solution, it shows how the algorithm affects the solution area in the changes in the number of surgeons, operating rooms and operating room reservations, which are among the operating room parameters. In the developed software, C# programming language has been preferred in order to provide comfortable use of the end user.

In this paper, a novel fuzzy fault tolerance algorithm is proposed for quarter vehicle active suspensions based on multi-objective constrained optimizations, which improves the riding comfort and driving safety. The objective function approach the optimal solutions via integrating multi-objective constraints as cost functions. By introducing the barrier function, the nonlinear active suspensions with multi-objective constraints are transformed into pure feedback systems without constraints. The problem is rather complicated yet challenging if the actuator faults are taken into account in quarter vehicle. The actuator faults are solved by the utilization of proportional actuation method. The unknown smooth dynamics based on physical truth are identified by exploiting the fuzzy logic knowledge. Meanwhile, the signals received in the suspensions do not violate the constraint boundary. Finally, the simulation results show that the algorithm is effective.

Supply chain network is important for the enterprise to improve the operation and management, but has become more complicated to optimize in reality. With the consideration of multiple objectives and constraints, this paper proposes a constrained large-scale multi-objective supply chain network (CLMSCN) optimization model. This model is to minimize the total operation cost (including the costs of production, transportation, and inventory) and to maximize the customer satisfaction under the capacity constraints. Besides, a coevolutionary algorithm based on the auxiliary population (CAAP) is proposed, which uses two populations to solve the CLMSCN problem. One population is to solve the original complex problem, and the other population is to solve the problem without any constraints. If the infeasible solutions are generated in the first population, a linear repair operator will be used to improve the feasibility of these solutions. To validate the effectivity of the CAAP algorithm, the experiment is conducted on the randomly generated instances with three different problem scales. The results show that the CAAP algorithm can outperform other compared algorithms, especially on the large-scale instances.

The Shuffled Complex Evolution—University of Arizona (SCE-UA) is a classical algorithm in the field of hydrology and water resources, but it cannot solve constrained optimization problems directly. Using penalty functions has been the preferred method to handle constraints, but the appropriate selection of penalty parameters and penalty functions can be challenging. To enhance the universality of the SCE-UA, we propose the Constrained Shuffled Complex Evolution Algorithm (CSCE) to conveniently and effectively solve inequality-constrained optimization problems. Its performance is compared with the SCE-UA using the adaptive penalty function (SCEA) on 14 test problems with inequality constraints. It is further compared with seven other algorithms on two test problems with low success rates. To demonstrate its effect in hydrologic model calibration, the CSCE is applied to the parameter optimization of the Xinanjiang (XAJ) model under synthetic data and observed data. The results indicate that the CSCE is more advantageous than the SCEA in terms of the success rate, stability, feasible rate, and convergence speed. It can guarantee the feasibility of the solution and avoid the problem of deep soil tension water capacity (WDM)

The Dual Active Bridge (DAB) DC-DC converter has the ability of bidirectional power transmission and the modulation scheme that is easy to implement, which can ensure the efficient transmission of energy in the system. Therefore, it is often used in various scenarios of small DC grid, such as energy storage, photovoltaic system, electric vehicle charging and so on. The main methods to improve the efficiency of dual active bridge include reducing the effective value of current and widening the soft switching area. Based on the above idea, an optimized extended phase-shift (EPS) modulation strategy is proposed in this paper. The modulation strategy achieves the goal of reducing the effective value of the current through the constraint optimization of the corresponding variables, and then improves the efficiency of the converter. In this paper, the working principles of several typical modulation strategies are introduced in detail, and then the power characteristics and soft-switching characteristics of the new method and other commonly used modulation schemes are analyzed and compared. Finally, the effectiveness of the optimization method for extended phase-shift modulation strategy is verified by the dual active bridge experimental prototype.

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.