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In combination with successful constraint handling techniques, a Matrix Adaptation Evolution Strategy (MA-ES) variant (the $\epsilon$MAg-ES) turned out to be a competitive algorithm on the constrained optimization problems proposed for the CEC 2018 competition on constrained single objective real-parameter optimization. A subsequent analysis points to additional potential in terms of robustness and solution quality. The consideration of a restart scheme and adjustments in the constraint handling techniques put this into effect and simplify the configuration. The resulting BP-$\epsilon$MAg-ES algorithm is applied to the constrained problems proposed for the IEEE CEC 2020 competition on Real-World Single-Objective Constrained optimization. The novel MA-ES variant realizes improvements over the original $\epsilon$MAg-ES in terms of feasibility and effectiveness on many of the real-world benchmarks. The BP-$\epsilon$MAg-ES realizes a feasibility rate of 100% on 44 out of 57 real-world problems and improves the best-known solution in 5 cases.

Several Variants of the Differential Evolution (DE) algorithm have been among the winners of competitions on bound constrained optimization problems in distinct editions of the Congress on Evolutionary Computation (CEC). This paper compares some of those variants presented in the last five years. Those algorithms are also combined in two adaptive dynamic Island Models (IM), which adjust the topology according to the algorithms applied in their islands. The Island Models are able to adaptively direct more solutions to the islands with more qualified algorithms to solve the problem. The results allow to verify whether the combination of the recently proposed DE variants in an adaptive dynamic structure is better than using each one individually.

One of the challenges that appear in solving constrained optimization problems is to quickly locate the search areas of interest. Although the initial solutions of any optimization algorithm have a significant effect on its performance, none of the existing initialization methods can provide direct information about the objective function and constraints of the problem to be solved. In this paper, a technique for generating initial solutions is proposed, which provides useful information about the behavior of both the objective function and the constraints. Based on such information, an automatic mechanism for selecting individuals, from the search areas of interest, is introduced. The proposed method is adopted with different evolutionary algorithms and tested on the CEC2006 and the CEC2010 test problems. The results obtained show the benefits of the proposed method in enhancing the performance, and reducing the average computational time, of several algorithms with respect to their versions adopting other initialization techniques

Connected and automated vehicles (CAVs) provide the most intriguing opportunity to reduce pollution, energy consumption, and travel delays. In earlier work, we addressed the optimal coordination of CAVs using Hamiltonian analysis. In this paper, we investigate the nature of the unconstrained problem and provide conditions under which the state and control constraints become active. We derive a closed-form analytical solution of the constrained optimization problem and evaluate the solution using numerical simulation.

We propose a novel projected gradient-like method to solve a minimization problem with simple constraints. The new search direction is consistent with the first-order optimality condition for the simple constrained optimization problem, and makes the search step-size easier to determine. Convergence properties of the method are analyzed. Numerical experiments are presented to illustrate the performance and application of the proposed method.

In this paper, a novel filled function method is proposed for solving constrained continuous global optimization problems without the coercive condition. The novel filled function, which is continuously differentiable and contains neither exponential term nor logarithmic term, is constructed at a point $x^{\ast}$ which is not necessarily a local optimal solution of the original problem. The implementation of the algorithm on several test problems is reported with numerical results.

PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment or blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale saddle-point system, which is challenging to solve, and acquiring a full space-time solution is often infeasible. We present a new framework to efficiently compute a low-rank approximation to the solution by reformulating the KKT system into a Sylvester-like matrix equation. This matrix equation is subsequently projected onto a small subspace via an iterative rational Krylov method and we obtain a reduced problem by imposing a Galerkin condition on its residual. In our work we discuss implementation details and dependence on the various problem parameters. Numerical experiments illustrate the performance of the new strategy also when compared to other low-rank approaches., 18 pages, 8 figures

The purpose of this paper is to evaluate the performance and computation time of Laguerre based model predictive control in the longitudinal motion of an aircraft. A Laguerre based model predictive algorithm is developed to control the longitudinal dynamics of an aircraft subject to state and input constraints. Then, the performance and the computation time of the designed controller are compared to the ones of the standard version of model predictive control. The results show that the Laguerre based model predictive control improves the computation time as much as 7.2% with a negligible trade-off in performance degradation. Because performing aircraft maneuvers needs to solve constrained optimization problems in short time samples, this study suggests the application of Laguerre based model predictive control for such purpose. Furthermore, by allowing to use shorter sampling times, the method could be used in real-time implementation of agile aircraft with fast dynamics.

The augmented Lagrange multiplier as an important concept in duality theory for optimization problems is extended in this paper to generalized augmented Lagrange multipliers by allowing a nonlinear support for the augmented perturbation function. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis. Meanwhile, the relations among generalized augmented Lagrange multipliers, saddle points, and zero duality gap property are developed.

Partial differential equation (PDE)–constrained optimization problems with control or state constraints are challenging from an analytical and numerical perspective. The combination of these constraints with a sparsity-promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an interior-point scheme applied to a smoothed reformulation of the discretized problem and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method, we introduce fast and efficient preconditioners that enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.

In order to evaluate efficiency of some global optimization method or compare efficiency of different methods, it is necessary to select a set of test problems, define comparison measures, and, finally, choose a way of visual presentation of the computational results. In this paper, a wide set of test optimization problems is considered including a new global constrained optimization problem generator (GCGen). Main performance measures and comparative criteria of efficiency are presented. The ways of visual presentation of computational results are suggested.

In 1988, Barzilai and Borwein published a pioneering paper which opened the way to inexpensively accelerate first-order. In more detail, in the framework of unconstrained optimization, Barzilai and Borwein developed two strategies to select the step length in gradient descent methods with the aim of encoding some second-order information of the problem without computing and/or employing the Hessian matrix of the objective function. Starting from these ideas, several effcient step length techniques have been suggested in the last decades in order to make gradient descent methods more and also more appealing for problems which handle large-scale data and require real- time solutions. Typically, these new step length selection rules have been tuned in the quadratic unconstrained framework for sweeping the spectrum of the Hessian matrix, and then applied also to nonquadratic constrained problems, without any substantial modification, by showing them to be very effiective anyway. In this paper, we deeply analyze how, in quadratic and nonquadratic mini- mization problems, the presence of a feasible region, expressed by a single linear equality constraint together with lower and upper bounds, inuences the spectral properties of the original Barzilai-Borwein (BB) rules, generalizing recent results provided for box-constrained quadratic problems. This analysis gives rise to modified BB approaches able not only to capture second-order informa- tion but also to exploit the nature of the feasible region. We show the benefits gained by the new step length rules on a set of test problems arising also from machine learning and image processing applications.

Retrieval networks are essential for searching and indexing. Compared to classification networks, attention visualization for retrieval networks is hardly studied. We formulate attention visualization as a constrained optimization problem. We leverage the unit L2-Norm constraint as an attention filter (L2-CAF) to localize attention in both classification and retrieval networks. Unlike recent literature, our approach requires neither architectural changes nor fine-tuning. Thus, a pre-trained network’s performance is never undermined.

In order to leverage the capacities of non-linear constraint solvers, we propose a reformulation of SAT into a box-constrained optimization problem where the objective function is polynomial. We prove that any optimal solution of the numerical problem corresponds to a solution of the Boolean formula, and demonstrate a stopping criterion that can be used with a numerical solver.

In this paper, we discuss necessary optimality conditions applicable to constrained optimization problems degenerating at the point of minimum. Some applications of these results to inverse function existence problem are studied.

We present a method for microtubule tracking in electron microscopy volumes. Our method first identifies a sparse set of voxels that likely belong to microtubules. Similar to prior work, we then enumerate potential edges between these voxels, which we represent in a candidate graph. Tracks of microtubules are found by selecting nodes and edges in the candidate graph by solving a constrained optimization problem incorporating biological priors on microtubule structure. For this, we present a novel integer linear programming formulation, which results in speed-ups of three orders of magnitude and an increase of 53% in accuracy compared to prior art (evaluated on three \(1.2\times 4\times 4\,\upmu \)m volumes of Drosophila neural tissue). We also propose a scheme to solve the optimization problem in a block-wise fashion, which allows distributed tracking and is necessary to process very large electron microscopy volumes. Finally, we release a benchmark dataset for microtubule tracking, here used for training, testing and validation, consisting of eight \(30 \times 1000 \times 1000\) voxel blocks (\(1.2\times 4\times 4\,\upmu \)m) of densely annotated microtubules in the CREMI data set (https://github.com/nilsec/micron).

Water alternating CO2 gas injection (WAG CO2) is one of the most promising enhanced oil recovery techniques. The optimization of this process requires performing many time‐consuming simulations. In this paper, an intelligent hybridization based on support vector regression (SVR) and genetic algorithm (GA) is introduced for the WAG process optimization in the presence of time‐dependent constraints. Multiple SVRs are used as dynamic proxy to mimic numerical simulator behavior in real time. Latin hypercube design (LHD) is applied to generate the proper runs to train the proxy and ten supplementary runs are randomly chosen to validate it. The goal of GA in this study is twofold. First, it is employed during the training of multiple SVRs to find their appropriate hyper‐parameters. Second, once the training and validation of the dynamic proxy are done, the GA is coupled with it to find the optimum WAG parameters which maximize field oil production total (FOPT) subject to time‐dependent water‐cut constraint and some domain constraints. The task is formulated as a non‐linear constrained optimization problem. A semi‐synthetic WAG CO2 case is used to examine the reliability of the approach. The results show that the established dynamic proxy is fast and accurate in reproducing the simulator outputs. The hybridization proxy‐GA is demonstrated to be reliable for the real‐time optimization of the formulated WAG process. © 2020 The Authors. Greenhouse Gases: Science and Technology published by Society of Chemical Industry and John Wiley & Sons, Ltd. © 2020 The Authors. Greenhouse Gases: Science and Technology published by Society of Chemical Industry and John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Subspace properties are presented for the trust-region subproblem that appears in the Augmented Lagrangian-Trust-Region method recently proposed by Wang and Yuan (2015). Specifically, when the approximate Lagrangian Hessians are updated by suitable quasi-Newton formulas, we show that any solution of the corresponding kth subproblem belongs to the subspace spanned by all gradient vectors of the objective and of the constraints computed up to iteration k. From this result, a subspace version of the referred method is proposed for large-scale equality constrained optimization problems. The subspace method is suitable to problems in which the number of constraints is much lower than the number of variables.

Applied research requires the usage of the proper statistics for hypothesis testing. Constrained optimization problems provide a framework that enables the researcher to build a statistic that fits his data and hypothesis at hand. In this paper I show some of the necessary conditions to obtain a Lagrange Multiplier test as well as some popular applications in order to highlight the usefulness of the test when the researcher must rely in asymptotic theory and to help the reader in the construction of a test in applied work.

The corridor allocation problem (CAP) seeks the optimum arrangement of given facilities along two sides of a central corridor. The CAP is so far handled as an unconstrained optimization problem without imposing any restriction on the placement of the facilities. In practice, however, some facilities may need to satisfy certain constraints on their placement. Accordingly, a constrained bi-objective CAP (cbCAP) model is proposed here, where some facilities are restricted to fixed, same and/or opposite rows. Realizing the difficulties to any algorithm for handling such a combinatorial problem, a cbCAP specific permutation-based genetic algorithm (cbCAP-pGA) with specialized operators is also proposed for solving the cbCAP model by generating only feasible solutions. In the numerical experimentation, the cbCAP-pGA is found capable in searching promising solutions even for a set of large-size benchmark instances.

In this chapter we study the convergence of the projected subgradient method for a class of constrained optimization problems in a Hilbert space. For this class of problems, an objective function is assumed to be convex but a set of admissible points is not necessarily convex. Our goal is to obtain an e-approximate solution in the presence of computational errors, where e is a given positive number.

Profile likelihood confidence intervals are a robust alternative to Wald's method if the asymptotic properties of the maximum likelihood estimator are not met. However, the constrained optimization problem defining profile likelihood confidence intervals can be difficult to solve in these situations, because the likelihood function may exhibit unfavorable properties. As a result, existing methods may be inefficient and yield misleading results. In this paper, we address this problem by computing profile likelihood confidence intervals via a trust-region approach, where steps computed based on local approximations are constrained to regions where these approximations are sufficiently precise. As our algorithm also accounts for numerical issues arising if the likelihood function is strongly non-linear or parameters are not estimable, the method is applicable in many scenarios where earlier approaches are shown to be unreliable. To demonstrate its potential in applications, we apply our algorithm to benchmark problems and compare it with 6 existing approaches to compute profile likelihood confidence intervals. Our algorithm consistently achieved higher success rates than any competitor while also being among the quickest methods. As our algorithm can be applied to compute both confidence intervals of parameters and model predictions, it is useful in a wide range of scenarios., Keywords: computer algorithm; constrained optimization; parameter estimation; estimability; identifiability

Distributed optimization algorithms contribute to an insight for the optimal coordination of distributed energy resources (DERs). To tackle the DER coordination problem, this paper studies a composite constrained optimization problem over multi-agent networks, where all agents independently maintain convex objective functions while satisfying the local and coupled constraints. Within the Peaceman-Rachford splitting (PRS) framework, a distributed algorithm based on the relaxed ADMM method is proposed for the considered problem, where agents in the network commonly share a fixed step-size instead of the diminishing one. Finally, the simulations on the optimal coordination of DERs including distributed generators (DGs) and energy storages (ESs) are conducted to validate the effectiveness of the proposed algorithm.

This paper introduces distributed adaptive algorithms for optimal step size selection in a distributed constrained optimization problem that involves stochastic target variations and noisy observations. The limit behavior of the step size sequences reflects fundamental impact that must be balanced between tracking the target changes and attenuating observation noises. Algorithms for simultaneously estimating target variation, tracking the global optimal solution, and finding the optimal step size are derived, which are shown to achieve convergence on all the sequences simultaneously to their respective optimal values.

The prediction of phase configurations and the optimization of operating parameters in the metallurgical process are normally achieved by the multiphase equilibrium calculation (MEC), which is formulated as a constrained optimization problem based on the principle of Gibbs free energy minimization. A revised quantum-behaved particle swarm optimization (RQPSO) algorithm has been proposed to solve the optimization problem using three-part improved strategies. Based on the KIVCET smelting characteristics, a MEC model for the KIVCET process is established and solved using the RQPSO algorithm. The calculated and industrial data of the lead grade are 96.20% and 96.33%, respectively, those of the matte grade are 19.39% and 19.68%, and the mass fractions of Pb in the predicted and industrial matte are 3.96% and 3.34%, respectively. The calculated results of the phase configuration are consistent with the actual production data, which indicates that the MEC model and RQPSO algorithm are accurate and reliable.

This paper presents a novel coupled similarity reference coding model (CSRC) by integrating non-negative constrained reference coding with coupled similarity metric for age-invariant face recognition (AIFR). Enhanced non-negative constrained reference coding is first proposed to encode the features into a meaningful part-based age-invariant representation, and we use iterative method combined with alternating direction method of multipliers to solve the troublesome constrained optimization problem. Coupled similarity metric is then proposed to eliminate the effect of irrelevant reference images in reference coding. By introducing non-negative constraint and coupled similarity metric, CSRC can obtain more meaningful representations and effectively enhance the performance of AIFR. Extensive experiments are performed on the CACD and MORPH datasets, which show an improvement in our model over state-of-the-art methods. What is more, experiment using deep features is performed and reaches high recognition rate, which shows that our model can be combined with deep networks to achieve better results.

An enhanced surrogate assisted framework, based on Probability of Improvement (PI) method, is proposed in this paper. We made some modifications to the original PI approach to enhance the performance of the modeling and optimization framework, leading to fewer rigorous simulations to find the optimal solution without loss of accuracy. We also extended the algorithm for handling general constraints using a fully probabilistic approach. The behavior of the proposed framework was investigated through a set of 9 Unconstrained Test Functions (UTF), 7 Constrained Optimization Problems (COP) and 3 Chemical Engineering Problems (CEP). The numerical results indicate that a lower number of rigorous model simulations were needed for optimizing UTF compared to the classic PI method and that the proposed framework was capable of achieving sustained near optimal solutions for COP and CEP. These results indicate that the proposed framework is suitable for solving computationally expensive constrained black-box optimization problems.

We study the charging control of plug-in electric vehicles in distribution networks with capacity constrained feeder lines. It is usually challenging to design an effective decentralized method to implement the optimal solution for the formulated constrained optimization problems due to the coupling relationship of charging behaviors on the constrained distribution network system. Alternatively, in this brief, in order to avoid possible overloading on the feeder lines, we propose a gradient projection-based decentralized method such that the step size is properly adjusted. We show that by applying the proposed method, the system converges and the implemented solution satisfies the capacity constraints of feeder lines. Simulation results show that the proposed method can effectively avoid possible overloads on the feeder lines in distribution networks.

In this paper, we provide an exact reformulation of Nonsmooth Constrained optimization Problems (NCP) using the Moreau-Yosida regularization. This reformulation allows the transformation of (NCP) to a sequence of convex programs of which solutions are feasible for (NCP). This sequence of solutions of auxiliary programs converges to a local solution of (NCP). Assuming Slater constraint qualification and basing on an exact penalization, our reformulation combined with a nonconvex proximal bundle method provides a local solution of (NCP). Our bundle method allows a strong update of the level set, may reduce significantly the number of null-steps and gives a new stopping criterion. Finally, numerical simulations are carried out.

In this paper, a new gravitation search algorithm with two masses is proposed to solve constrained optimization problems. In this algorithm, each particle has two masses, namely, "feasible mass" and "infeasible mass". If the particle in the feasible region is updated, then the feasible mass is used in gravitation search algorithm, otherwise, the infeasible mass is used. This idea is based on that a feasible particle is attracted towards better feasible particles and an infeasible particle is attracted towards the feasible region. Finally, the proposed algorithm is tested on 10 benchmark problems and compared with the other algorithms. The numerical results indicate that the proposed algorithm has the better performance in solving constrained optimization problems.

To apply MOEAs to industrial problems, the estimation of the ideal and nadir points becomes crucial in order to handle the differently scaled objectives by normalization and to handle the unknown Pareto Front. In this paper, the impact of the estimation methods of the ideal and nadir points for decomposition-based multi-objective evolutionary algorithms is examined on some constrained optimization problems whose constraints are explicitly designed to have the characteristics of the practical problems. The numerical experiments show that the estimation method has great impact on the performance of MOEAs.

Digital microfluidic biochips (DMFB) are a part of lab-on-a-chip (LOC) devices. DMFBs are designed to carry out biochemical analysis efficiently. Scheduling DMFB operations is a constrained optimization problem, and the respective decision problem is NP-complete. We propose a hybrid artificial bee colony (ABC) algorithm using generalized N-point crossover (GNX) based Scheduling for DMFB operations. There are several schedules possible for a given set of microfluidic operations. Heuristic algorithms are capable of finding one such possible schedule from search space (possible schedules) while our proposed ABC-GNX perturbs through search space, evaluates several possible schedules, and outputs optimal schedule among the evaluated schedules. While several iterative improvement based algorithms exist, which can do the same but, the proposed ABC-GNX algorithm finds the optimal solution in less execution time as compared to existing algorithms. ABC-GNX produces a higher number of optimal assay completion times and less execution time on evaluated benchmarks.

This paper concerns a nonsmooth sparsity constrained optimization problem. We present first and second-order necessary and sufficient optimality conditions by using the concept of normal and tangent cones to the sparsity constraint set. Moreover, second-order tangent set to the sparsity constraint is described and then a new second-order necessary optimality condition is established. The results are illustrated by several examples.

This paper presents the application of a novel algorithm, which is based on a modified Sine-Cosine technique for solving the optimal power flow (OPF) problem. It is a highly coupled non-linear constrained optimization problem. The modified Sine-Cosine algorithm (MSCA) aims at reducing the computational time with a sufficient improvement in finding the optimal solution and feasibility. Levy flights are added to the original Sine-Cosine algorithm to enhance the ability to focus on optimal and avoids local optima. Decreasing search agents number based on the objective function speeds up the MSCA. The MSCA presents a simple and robust solution for the OPF problem under different objective functions. The MSCA is validated with solving the OPF problem for a number of benchmark test systems, namely the IEEE-30 bus and IEEE 118-bus systems under selected objective functions, which are based on operational and economic performance indices of the power system. The proposed MSCA is compared with other optimization methods to illustrate the effectiveness and potential of the SCA and MSCA algorithms.

Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? We establish a quantitative answer to this question for exponents $q$ sufficiently close to even integers, building on previous work proving the existence of such maximizers., 34 pages

Numerical methods for computing plausibility and belief distributions of consequences of a subjective model are considered. More precisely, related constrained optimization problems are studied. Error estimates of the proposed algorithms are obtained. Techniques for taking into account the information about the consequence available to the researcher for improving the accuracy of computations are discussed.

Artificial Bee Colony (ABC) is one of the most popular nature inspired optimization algorithms. Recently, a variant of ABC, Gbest-guided ABC (GABC) was proposed. GABC was verified to perform better than ABC, in terms of efficiency and reliability. In the position update process of GABC, Gbest (the best individual in the swarm) individual influences the movement of the swarm. This movement may create a cluster around the Gbest individual which further leads to the premature convergence, particularly for constrained optimization problems. This paper presents a modification in GABC for constrained optimization problems. GABC is modified in both employed and onlooker bee phases by incorporating the concept of fitness probability based individual movement. The modified GABC is tested over 20 constrained benchmark problems and applied to solve 3 engineering design problems. Optimal power flow problem has also been solved using modified GABC to check the efficiency of the proposed algorithm.

This paper gives details on how to obtain mathematically rigorous results for global unconstrained and equality constrained optimization problems, as well as for finding all roots of a nonlinear fu...

This article analyses the influence of Robbins’s thought on the subsequent development of economics imperialism. First, the analysis of some passages that have been overlooked up to the present, in which the author explicitly expresses his disbelief in the omnipotence of the economic method, makes it possible to argue that Robbins did not share the confidence exhibited by the advocates of economics imperialism in the capacity of economics to explain non-economic phenomena. Second, this article shows that Robbins describes the influence of real scarcity on human behaviour in terms of a (static) constrained maximization problem, thereby confusing issues of method and scope in his writings. This confusion facilitated the view that economics is a method without a proper subject matter and its later expansion into other fields. In conclusion, although Robbins cannot be counted as an earlier promoter of economics imperialism, the misunderstandings implicit in his writings paved the way for the emergence of this intellectual movement.

Empirical likelihood (EL) is a nonparametric method based on observations. EL method is defined as a constrained optimization problem. The solution of this constrained optimization problem ...

Optimal control problems for PDEs arise in many important applications. A main step in the solution process is the solution of the arising linear system, where the crucial point is usually finding a proper preconditioner. We propose both proper block diagonal and more involved preconditioners, and derive mesh independent superlinear convergence of the preconditioned GMRES iterations based on a compact perturbation property of the underlying operators.

This paper presents a hybrid algorithm based on using moth-flame optimization (MFO) algorithm with simulated annealing (SA), namely (SA-MFO). The proposed SA-MFO algorithm takes the advantages of both algorithms. It takes the ability to escape from local optima mechanism of SA and fast searching and learning mechanism for guiding the generation of candidate solutions of MFO. The proposed SA-MFO algorithm is applied on 23 unconstrained benchmark functions and four well-known constrained engineering problems. The experimental results show the superiority of the proposed algorithm. Moreover, the performance of SA-MFO is compared with well-known and recent meta-heuristic algorithms. The results show competitive results of SA-MFO concerning MFO and other meta-heuristic algorithms.

This article deals with the Tikhonov regularization method for the constrained Lagrange approach, taking into account polylinear programming problems. A regularized Lagrange function is str...

Firefly and cuckoo search algorithms are two of the most widely used nature-inspired algorithms due to their simplicity and inexpensive computational cost when they applied to solve a wide range of problems. In this article, a new hybrid algorithm is suggested by combining the firefly algorithm and the cuckoo search algorithm to solve constrained optimization problems (COPs) and real-world engineering optimization problems. The proposed algorithm is called Hybrid FireFly Algorithm and Cuckoo Search (HFFACS) algorithm. In the HFFACS algorithm, a balance between the exploration and the exploitation processes is considered. The main drawback of the firefly algorithm is it is easy to fall into stagnation when the new solution is not better than its previous best solution for several generations. In order to avoid this problem, the cuckoo search with Levy flight is invoked to force the firefly algorithm to escape from stagnation and to avoid premature convergence. The proposed algorithm is applied to six benchmark constrained optimization problems and five engineering optimization problems and compared against four algorithms to investigate its performance. The numerical experimental results show the proposed algorithm is a promising algorithm and can obtain the optimal or near optimal solution within a reasonable time.

A joint chance constrained optimization problem involves multiple uncertain constraints, i.e., constraints with stochastic parameters, that are jointly required to be satisfied with probability exc...

In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.

A new method for solving the inequality constrained optimization problem is proposed for the case when the system of necessary optimality conditions of Kuhn—Tucker is degenerate. This situation occurs for example in the case when strict complementarity conditions fails in solution point. The reduction of the inequalities con- strained optimization problem to the equalities constrained problem is substantiated and the use of a new 2-fac- tor Newton method for the effective solution of the obtained degenerate system of optimality conditions is shown.