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In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

We focus on optimality conditions for an important class of nonconvex and nonsmooth optimization problems, where the objective and constraint functions are presented as a difference of two tangentially convex functions. The main contribution of this paper is to clarify several kinds of stationary solutions and their relations, and establish local optimality conditions with a nonconvex feasible set. Finally, several examples are given to illustrate the effectiveness of the obtained results.

In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

The rapidly changing requirements of engineering optimization problems require unprecedented levels of compatibility to integrate diverse uncertainty information to search optimum among design region. The sophisticated optimization methods tackling uncertainty involve reliability-based design optimization and robust design optimization. In this paper, a novel alternative approach called risk-based design optimization (RiDO) has been proposed to counterpoise design results and costs under hybrid uncertainties. In this approach, the conditional value at risk (CVaR) is adopted for quantification of the hybrid uncertainties. Then, a CVaR estimation method based on Monte Carlo simulation (MCS) scenario generation approach is derived to measure the risk levels of the objective and constraint functions. The RiDO under hybrid uncertainties is established and leveraged to determine the optimal scheme which satisfies the risk requirement. Three examples with different calculation complexity are provided to verify the developed approach.

In this paper, we consider the set D of inequalities with nonconvex constraint functions in the face of data uncertainty. We show under a suitable condition that “perturbation property” of the robust best approximation to any $$x\in {\mathbb {R}}^n$$ from the set $$\tilde{K}:={{\bar{C}}} \cap D$$ is characterized by the strong conical hull intersection property (strong CHIP) of $${\bar{C}}$$ and D. The set C is an open convex subset of $${\mathbb {R}}^n$$ and the set D is represented by $$D:=\{x\in {\mathbb {R}}^n: g_{j}(x,v_j)\le 0, \; \forall \; v_j\in V_j, \; j=1,2,\ldots ,m\},$$ where the functions $$g_j:{\mathbb {R}}^n\times V_j\longrightarrow {\mathbb {R}}, \; j=1,2,\ldots ,m,$$ are continuously Frechet differentiable that are not necessarily convex, and $$v_j$$ is the uncertain parameter which belongs to an uncertainty set $$V_j\subset {\mathbb {R}}^{q_j}, \; j=1,2,\ldots ,m.$$ This is done by first proving a dual cone characterization of the robust constraint set D. Finally, following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty under the robust nondegeneracy constraint qualification. Given examples illustrate the nature of our assumptions.

We develop a unified level-bundle method, called accelerated constrained level-bundle (ACLB) algorithm, for solving constrained convex optimization problems. where the objective and constraint functions can be nonsmooth, weakly smooth, and/or smooth. ACLB employs Nesterov’s accelerated gradient technique, and hence retains the iteration complexity as that of existing bundle-type methods if the objective or one of the constraint functions is nonsmooth. More importantly, ACLB can significantly reduce iteration complexity when the objective and all constraints are (weakly) smooth. In addition, if the objective contains a nonsmooth component which can be written as a specific form of maximum, we show that the iteration complexity of this component can be much lower than that for general nonsmooth objective function. Numerical results demonstrate the effectiveness of the proposed algorithm.

In this paper, the sine-cosine optimization algorithm (SCO) is used to solve the shape optimization of a vehicle clutch lever. The design problem is posed for the shape optimization of a clutch lever with a mass objective function and a stress constraint. Actual function evaluations are based on finite element analysis, while the response surface method is used to obtain the equations for objective and constraint functions. Recent optimization techniques such as the salp swarm algorithm, grasshopper optimization algorithm, and sine-cosine algorithm are used for shape optimization. The results show the ability of the sine-cosine optimization algorithm to optimize automobile components in the industry. King Fahd University of Petroleum and Minerals Khon Kaen University

Structural optimization software that can produce high-resolution designs optimized for arbitrary cost and constraint functions is essential to solve real-world engineering problems. Such requirements are not easily met due to the large-scale simulations and software engineering they entail. In this paper, we present a large-scale topology optimization framework with adaptive mesh refinement (AMR) applied to stress-constrained problems. AMR allows us to save computational resources by refining regions of the domain to increase the design resolution and simulation accuracy, leaving void regions coarse. We discuss the challenges necessary to resolve such large-scale problems with AMR, namely, the need for a regularization method that works across different mesh resolutions in a parallel environment and efficient iterative solvers. Furthermore, the optimization algorithm needs to be implemented with the same discretization that is used to represent the design field. To show the efficacy and versatility of our framework, we minimize the mass of a three-dimensional L-bracket subject to a maximum stress constraint and maximize the efficiency of a three-dimensional compliant mechanism subject to a maximum stress constraint.

In this article, we use the robust optimization approach (also called the worst-case approach) for finding e-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions. Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an e-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions.

Reliability-based design optimization (RBDO) is often associated with computationally expensive computer simulations in practical engineering applications, in which both objective and constraint functions may be implicit and time-consuming to evaluate. To improve the efficiency of RBDO, metamodels are often employed to substitute the implicit and expensive functions. In this paper, a new local update-based method is proposed for RBDO under the approximation of both objective and constraint functions. In this method, Kriging metamodels are employed to replace both objective and constraint functions, and sequential optimization and reliability assessment (SORA) is adopted to perform RBDO. Meanwhile, a local update method is developed for the approximation of objective and constraint functions, termed as LUOC. In LUOC, two local update strategies are proposed. For the constraint functions, one local update strategy judges the active constraints and then sequentially refines their Kriging metamodels in the local regions around the most probable target points obtained by SORA. For the objective function, the other local update strategy based on expected improvement is developed to refine the Kriging metamodel in the local region around the SORA solution with considering the Kriging prediction variance. Three examples including a battery design application are presented to test the accuracy and efficiency of the proposed method for RBDO.

This article presents an implementation of one of the latest optimization methods of obtaining light vehicle designs. First, the problem of coupling with a bolted rim is optimized using the butterfly optimization algorithm (BOA). Finally, the BOA is used to solve the shape optimization of a vehicle suspension arm. It is utilized from the Kriging metamodeling method to obtain equations of objective and constraint functions in shape optimization. At the end of the research effort in this paper, the weight reduction of the suspension arm by using the BOA is 32.9%. The results show the BOA's ability to design better optimum components in the automotive industry. King Fahd University of Petroleum and Minerals Khon Kaen University

In this paper, zero duality gap conditions in nonconvex optimization are investigated. It is considered that dual problems can be constructed with respect to the weak conjugate functions, and/or directly by using an augmented Lagrangian formulation. Both of these approaches and the related strong duality theorems are studied and compared in this paper. By using the weak conjugate functions approach, special cases related to the optimization problems with equality and inequality constraints are studied and the zero duality gap conditions in terms of objective and constraint functions, are established. Illustrative examples are provided.

Holant problems are a family of counting problems parameterised by sets of algebraic-complex valued constraint functions, and defined on graphs. They arise from the theory of holographic algorithms, which was originally inspired by concepts from quantum computation. Here, we employ quantum information theory to explain existing results about holant problems in a concise way and to derive two new dichotomies: one for a new family of problems, which we call Holant$^+$, and, building on this, a full dichotomy for Holant$^c$. These two families of holant problems assume the availability of certain unary constraint functions -- the two pinning functions in the case of Holant$^c$, and four functions in the case of Holant$^+$ -- and allow arbitrary sets of algebraic-complex valued constraint functions otherwise. The dichotomy for Holant$^+$ also applies when inputs are restricted to instances defined on planar graphs. In proving these complexity classifications, we derive an original result about entangled quantum states., 57 pages, combines edited versions of arXiv:1702.00767 and arXiv:1704.05798 with some new results

This paper presents a novel Generative Neural Network Architecture for modelling the inverse function of an Artificial Neural Network (ANN) either completely or partially. Modelling the complete inverse function of an ANN involves generating the values of all features that corresponds to a desired output. On the other hand, partially modelling the inverse function means generating the values of a subset of features and fixing the remaining feature values. The feature set generation is a critical step for artificial neural networks, useful in several practical applications in engineering and science. The proposed Oracle Guided Generative Neural Network, dubbed as OGGN, is flexible to handle a variety of feature generation problems. In general, an ANN is able to predict the target values based on given feature vectors. The OGGN architecture enables to generate feature vectors given the predetermined target values of an ANN. When generated feature vectors are fed to the forward ANN, the target value predicted by ANN will be close to the predetermined target values. Therefore, the OGGN architecture is able to map, inverse function of the function represented by forward ANN. Besides, there is another important contribution of this work. This paper also introduces a new class of functions, defined as constraint functions. The constraint functions enable a neural network to investigate a given local space for a longer period of time. Thus, enabling to find a local optimum of the loss function apart from just being able to find the global optimum. OGGN can also be adapted to solve a system of polynomial equations in many variables. The experiments on synthetic datasets validate the effectiveness of OGGN on various use cases.

This note corrects an error in our paper [RAIRO: OR 55 (2021) 1–11] as we should drop the expression “with at least one strict inequality” in the definition of interval order in Section 2. Instead of proposing this short amendment, the authors of [RAIRO: OR 55 (2021) 13–22] gave a proposition that requires an additional condition on the constraint functions. However, we claim that all the results of our paper are correct once the modification above is done.

We introduce robust weak sharp and robust sharp solution to a convex programming with the objective and constraint functions involved uncertainty. The characterizations of the sets of all the robust weak sharp solutions are obtained by means of subdiferentials of convex functions, DC fuctions, Fermat rule and the robust-type subdifferential constraint qualification, which was introduced in X.K. Sun, Z.Y. Peng and X. Le Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim Lett. 10. (2016), 1463-1478. In addition, some applications to the multi-objective case are presented.

Scheduling workers is one of the problems faced by every company. The regulations set by the company, the availability of the number of workers, and the division of labor are the determining factors in the scheduling system. This worker scheduling problem can be modeled as an Integer Programming problem. Integer Programming is an optimization technique with linear objective functions, linear constraint functions, and integer variables. This paper discusses the formulation of worker scheduling problems in the form of Integer Programming with workers in companies engaged in the production of Crumb Rubber with the objective function of minimizing the number of workers employed. The next model is implemented using the help of LINGO 11.0 software. The implementation results show that the model is able to produce optimal employee schedules.

Over two decades, the fractional (non-integer) order PID (FOPID or PIλDµ ) controller was introduced and demonstrated to perform the better responses in comparison with the conventional integer order PID (IOPID). In this paper, the design of an optimal FOPID controller for a DC motor speed control system by the flower pollination algorithm (FPA), oneof the most efficient population-based metaheuristic optimization searching techniques, is proposed. Based on the modern optimization framework, five parameters of the FOPID controller are optimized by the FPA to meet the response specifications of the DC motor speed control system and defined as constraint functions. Results obtained by the FOPID controller are compared with those obtained by the IOPID designed by the FPA. As the simulation results show, the FOPID can provide significantly superior speed responses to the IOPID.

In this paper, we establish the primal and dual second-order necessary and sufficient optimality conditions for (local) weakly efficient solutions of vector equilibrium problems with constraints in terms of contingent derivatives with 2-steady functions in real finite spaces. Using the 2-steadiness of objective and constraint functions at a given optimal point, the second-order necessary conditions for weakly efficient solutions of constrained vector equilibrium problems are derived. Under the suitable assumptions on 2-steadiness of objective and constraint functions, the dual second-order necessary optimality conditions will become the dual second-order sufficient optimality conditions. We also give several examples that illustrate our results.

This article proposes a new algorithm for computing solutions to mixed flow problems involving the classical Lighthill-Whitham-Richards (LWR) flow model. We show that the behavior of autonomous vehicles can be described by viability constrained solutions to an alternate Hamilton-Jacobi formulation of the same model, for appropriate upper constraint functions. We also provide the physical interpretation of viability constrained solutions to the model. This results in a viability-based framework that can be used to incorporate the effect of autonomous vehicles in traffic, even when these vehicles are allowed to switch between modes (autonomous/non autonomous). We then illustrate this algorithm on realistic traffic simulation examples involving partial switched autonomous vehicle flows.

In many optimization problems arising from scientific, engineering and artificial intelligence applications, objective and constraint functions are available only as the output of a black-box or simulation oracle that does not provide derivative information. Such settings necessitate the use of methods for derivative-free, or zeroth-order, optimization. We provide a review and perspectives on developments in these methods, with an emphasis on highlighting recent developments and on unifying treatment of such problems in the non-linear optimization and machine learning literature. We categorize methods based on assumed properties of the black-box functions, as well as features of the methods. We first overview the primary setting of deterministic methods applied to unconstrained, non-convex optimization problems where the objective function is defined by a deterministic black-box oracle. We then discuss developments in randomized methods, methods that assume some additional structure about the objective (including convexity, separability and general non-smooth compositions), methods for problems where the output of the black-box oracle is stochastic, and methods for handling different types of constraints.

The aim of this paper is to establish the Kuhn–Tucker optimality conditions for the efficient solution types of constrained vector equilibrium problems in terms of directional derivatives in Banach spaces. Under the suitable assumptions on generalized convexity of objective and constraint functions, the Kuhn–Tucker necessary and sufficient optimality conditions for efficient solution, weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem with set and cone constraints are established. Some applications to the constrained vector variational inequality problem and the constrained vector optimization problem are also given. Besides, the Karush–Kuhn–Tucker necessary and sufficient optimality conditions for weakly efficient solutions to the model of transportation–production and Nash–Cournot equilibria problems are obtained. We also provide several examples to illustrate our results.

This paper proposes a method for the solution of constrained min-max problems. The method is tested on a benchmark of representative problems presenting different structures for the objective function and the constraints. The particular min-max problem addressed in this paper finds application in optimisation under uncertainty when the constraints need to be satisfied for all possible realisations of the uncertain quantities. Hence, the algorithm proposed in this paper search for solutions that minimise the worst possible outcome for the objective function due to the uncertainty while satisfying the constraint functions in all possible scenarios. A constraint relaxation and a scalarisation procedure are also introduced to trade-off between objective optimality and constraint satisfaction when no feasible solutions can be found.

This paper is devoted to the investigation of a nonsmooth multiobjective bilevel programming problem with equilibrium constraints ((MBPP) for short) in terms of convexificators in finite-dimensional spaces. We present necessary optimality conditions for the local weak efficient solution to such problem. Under the Mangasarian–Fromovitz and generalized standard Abadie type constraint qualification in the sense of convexificators, we establish as an application the Wolfe and Mond-Weir type dual problem for the problem (MBPP). Besides, we provide strong and weak duality theorems for the original problem and its Wolfe and Mond–Weir type dual problem under suitable assumptions on the $$\partial ^*$$ -convexity and the upper semi-regularity of objective and constraint functions. Illustrative examples are also proposed to demonstrate the main results of the paper.

In the past years, a significant amount of research has been done in optimizing computationally expensive and time-consuming objective functions using various surrogate modeling approaches. Constraints have often been neglected or assumed to be a by-product of the expensive objective computation and thereby being available after executing the expensive evaluation routines. However, many optimization problems in practice have separately evaluable computationally inexpensive geometrical or physical constraint functions, while the objectives may still be time-consuming. This scenario probably makes the simplest case of handling heterogeneous and multi-scale surrogate modeling in the presence of constraints. In this paper, we propose a method which makes use of the inexpensiveness of constraints to ensure all time-consuming objective evaluations are only executed for feasible solutions. Results on test and real-world problems indicate that the proposed approach finds a widely distributed set of near-Pareto-optimal solutions with a small budget of expensive evaluations.

The paper proposes and substantiates the use of the method of correction functions in modeling the processes of corrosion deformation in structures operating in aggressive environments. This approach is especially relevant for optimal design problems, when the calculation of the constraint functions involves modeling this process. In this method a solution to the durability problem corresponding to a specific value of the vector of variable parameters is sought as a product of two functions: an approximate solution of the problem, and a correction function (error of the approximate solution). An approximate solution is sought for a simplified model of corrosion deformation of a structure, i.e. some of data on which a solution depends is ignored. This data is used to construct the correction function. On the other hand, the correction function doesn’t depend on data determining design parameters. Therefore, it is invariant with respect to the constructions of the considered class. The paper proposes to consider the correction function as a neural network with a single output neuron - the error of an approximate solution. The calculation of the constraint functions, thus, implies an approximate solution to the problem, which can be obtained with minimal computational costs after refinement of a result. It is shown that the joint use of the computational methods of solid mechanics and computational intelligence methods (including artificial neural networks and genetic methods) will significantly improve the quality of solving complex applied problems in the mechanics of corroding structures.

The augmented Lagrangian method is a classical solution method for nonlinear optimization problems. At each iteration, it minimizes an augmented Lagrangian function that consists of the constraint functions and the corresponding Lagrange multipliers. If the Lagrange multipliers in the augmented Lagrangian function are close to the exact Lagrange multipliers at an optimal solution, the method converges steadily. Since the conventional augmented Lagrangian method uses inaccurate estimated Lagrange multipliers, it sometimes converges slowly. In this paper, we propose a novel augmented Lagrangian method that allows the augmented Lagrangian function and its minimization problem to have variable constraints at each iteration. This allowance enables the new method to get more accurate estimated Lagrange multipliers by exploiting Karush–Kuhn–Tucker points of the subproblems and consequently to converge more efficiently and steadily.

In the context of real-time optimization, modifier-adaptation schemes update the model-based optimization problem by adding first-order correction terms to the cost and constraint functions of the optimization problem. This guarantees meeting the plant first-order optimality conditions upon convergence, despite the presence of parametric and structural plant-model mismatch. An alternative modifier-adaptation strategy has been proposed, wherein the first-order corrections are applied to the output functions rather than to the cost and constraint functions. This paper analyzes this alternative adaptation scheme in detail and provides arguments for its use in cases where the cost and constraints are nonlinear functions of the inputs and outputs. The approach is illustrated in simulation on the Williams–Otto continuous stirred-tank reactor.

This work deals with the optimum design of variable angle-tow (VAT) composites in the theoretical framework of B-Spline surfaces. The design problem is formulated as a constrained optimisation problem in the context of the multi-scale two-level (MS2L) optimisation methodology. The proposed approach aims at optimising all parameters characterising the VAT laminate at each pertinent scale (mesoscopic and macroscopic ones). In particular, this study focuses on the first level of the MS2L optimisation strategy that aims at determining, at the VAT laminate macroscopic scale, the optimum distribution of the laminate mechanical properties over the structure. At this stage, the behaviour of the VAT composite is described by means of the polar formalism and the spatial distribution of the polar parameters is represented through a set of suitable B-Spline surfaces. The expression of the gradient of objective and constraint functions is determined in a closed form by exploiting the properties of the B-Spline surfaces. Moreover, the effect of the discrete variables tuning the shape of the B-Spline surfaces (mainly the number of control points) on the quality of the optimum solution is investigated. The effectiveness of the proposed approach is tested through some meaningful benchmarks taken from literature.

This paper provides some new results on robust approximate optimal solutions of a fractional semi-infinite optimization problem under uncertainty data in the constraint functions. By employing conjugate analysis and robust optimization approach (worst-case approach), we obtain some necessary and sufficient optimality conditions for robust approximate optimal solutions of such a fractional semi-infinite optimization problem. In addition, we state a mixed type approximate dual problem to the reference problem and obtain some robust duality properties between them. The results obtained in this paper improve the corresponding results in the literature.

This paper is devoted to constructing Wolfe and Mond–Weir dual models for interval-valued pseudoconvex optimization problem with equilibrium constraints, as well as providing weak and strong duality theorems for the same using the notion of contingent epiderivatives with pseudoconvex functions in real Banach spaces. First, we introduce the Mangasarian–Fromovitz type regularity condition and the two Wolfe and Mond–Weir dual models to such problem. Second, under suitable assumptions on the pseudoconvexity of objective and constraint functions, weak and strong duality theorems for the interval-valued pseudoconvex optimization problem with equilibrium constraints and its Mond–Weir and Wolfe dual problems are derived. An application of the obtained results for the GA-stationary vector to such interval-valued pseudoconvex optimization problem on sufficient optimality is presented. We also give several examples that illustrate our results in the paper.

In this paper, we present a quadratic model for minimizing problems with nonconvex and nonsmooth objective and constraint functions. This method is based on sequential quadratic programming...

Whereas topology optimization has achieved immense success, it involves an intrinsic difficulty. That is, optimized structures obtained by topology optimization strongly depend on the settings of the objective and constraint functions, i.e., the formulation. Nevertheless, the appropriate formulation is not usually obvious when considering structural design problems. Although trial-and-error to determine appropriate formulations are implicitly performed in several studies on topology optimization, it is important to explicitly support the process of trial-and-error. Therefore, in this study, we propose a new framework for topology optimization to determine appropriate formulations. The basic idea of this framework is incorporating knowledge discovery in databases (KDD) and topology optimization. Thus, we construct a database by collecting various and numerous material distributions that are obtained by solving various structural design problems with topology optimization, and find useful knowledge with respect to appropriate formulations from the database on the basis of KDD. An issue must be resolved when realizing the above idea, namely the material distribution in the design domain of a data record must be converted to conform to the design domain of the target design problem wherein an appropriate formulation should be determined. For this purpose, we also propose a material distribution-converting method termed as design domain mapping (DDM). Several numerical examples are used to demonstrate that the proposed framework including DDM successfully and explicitly supports the process of trial-and-error to determine the appropriate formulation.

Hedef Programlamada karar verici tarafından belirlenen hedeflerin arzu edilen seviyeleri eğer gerçekçi değil ise, çözümün sonucunda hedeflerden sapmalar çok yüksek değerde gerçekleşebilir. Bu durum karar verici yanlış karar almasına yol açar. Benzer durum Bulanık Hedef programlama için de geçerlidir. Çünkü bulanık hedefler ve bu hedeflerin tolerans değerleri doğru belirlenmezse hedeflerden sapmalar artacaktır. Ayrıca ister Hedef Programlama problemi ister Bulanık Hedef programlama probleminde eğer hedeflerin yanı sıra kısıt fonksiyonları da olursa, kısıtlara bağlı çözüm gerçekleşeceği için doğru tanımlanmamış hedef değerlerden sapmalar çok fazla olacaktır. Çünkü hedefler kısıtlar tarafından sınırlandırılır. Bu çalışmada Çok Amaçlı Doğrusal Programlama modelinde kurulan el yapımı mobilya üretimi yapan bir işletme problemi Bulanık Hedef Programlama modelinde çözülebilmesi için ilk olarak amaç fonksiyonlarının pozitif ve negatif ideal çözümleri belirlenmiştir. Daha sonra her bir amaç fonksiyonu pozitif ve negatif ideal çözümler kullanılarak bulanık hedeflere dönüştürülmüştür.

We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the $p$-norm function, where $p$ is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in [O. L. Mangasarian, Absolute value programming, Computational Optimization and Applications 36 (2007) pp. 43-53]. In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems., 21 pages

PT.Wulantika Utama is one of distributor in Palu who distribute products from factories to retailers. The purpose of this research are to maximize the distribution numbers of trucks and minimize the cost of distribution. Goal Programming is a method that can solve the problem with more than one purpose. Goal Programming model formulation in this study consists of 5 priority and 5 constraint functions. The fifth priority is the storage capacity, the number of trucks used for the distribution of goods to Donggala, Ampana, Poso, and cost targets are minimum distribution. Constraint function consists of inventory in the warehouse, the number of trucks and distribution costs. The results showed that the supply of goods by the warehouse capacity that can fulfiil the necessary distribution product during a month amounted to 14.755 cartons. Optimal distribution for each destination, in Donggala are 8 trucks with a capacity of 250 cartons, Poso are 20 trucks with a capacity of 300 cartons and Ampana are 14 trucks with a capacity of 500 cartons. This result can save distribution costs of Rp.90.993.009 from the previous distribution costs of Rp. 282 602.689.

PT.Sinar Niaga Sejahtera isone ofdistributorin Palu who distribute products to a variety of shops. Goal Programming is a method that can solve the problem with more than one purposes. The purposes of this study are to maximize the number of the car and minimize the distribution cost of PT.Sinar Niaga Sejahtera. Goal Programming model formulationin this research consistsof 6 priorities and 6 function constraints. The sixth priorities are,warehous capacity, the number of cars used to the distribution of goods to store Sinar Kasih II,store Cahaya Indah, store Bintang Rezeki, store Hi. Abdullah, and a minimum distribution costs. Constraint functions consist of a number of cars and the cost of distribution. The research results showed that the supply of goods by the warehouse capacity that can fulfiil the necessary distribution of goods during the month amounted to 136.93 or 8.628 box Optimal volume distribution of goods in each store are sequentially Sinar Kasih II which is 2 units with a capacity of 4 or 252 box, Cahaya Indah 3 units with a capacity of 7 or 441 box, Hi. Abdullah 2 units with a capacity of 12 or 756 box and Star 2 cars Rezeki capacity of 4 . This model can save the distribution costs of Rp. 7.127.147 from the previous distribution costs of Rp. 35.000.000.

In this paper, we consider a class of time-delay optimal control problem (TDOCP) with canonical equality and inequality constraints. By applying control parameterization method together with time-scaling transformation, a TDOCP can be readily solved by gradient-based optimization methods. The partial derivative of the cost as well as the constraint functions with respect to the decision variables are obtained by variational approach, which is inefficient when the discretization for the control function is relatively dense. For general optimal control problem without time-delay, co-state approach is an effective way to compute the gradients, however, when time-delay is involved in the dynamic system, the co-state system is not known. In this paper, we derive the co-state system for TDOCP to compute the gradients of the cost and constraints. Numerical results show that the computational efficiency is much higher when compared with the traditional variational approach.

The product design process has a very important effect on product costs. It aims to develop products that are able to compete by achieving an optimum design in the product design process. This research presents the first application in the literature of the grey wolf, whale, water cycle, ant lion and sine-cosine optimization algorithms for the optimum design of vehicle components. In this study, the optimal structural model of a vehicle connecting rod was determined. In the optimization process, various design alternatives were created by using the latin hypercube method. Stress analysis was performed for each of these designs. According to the generated responses, equations for objective and constraint functions were obtained. The optimization problem was solved using the above mentioned algorithms which have been newly developed in the literature, resulting in optimum connection rod design. The results demonstrate that the grey wolf, whale, water cycle, ant lion and sine-cosine algorithms are very important options in optimizing design and manufacturing optimization problems.

Previous behavioral research indicates that native speakers of Chinese are not very sensitive to lexical gender information, and one question that remains is how the rule of gender constraint functions in L1 and L2 reflexive pronoun resolution by Chinese-English bilinguals. To answer the question, the present study conducted two ERP experiments in which participants' ERP responses to L1 and L2 reflexive pronouns that were either congruent or incongruent with their antecedents were recorded and analyzed. Gender incongruity effect was observed in the L1 in the form of left anterior negativity effect (LAN) within 350–450 ms, but in the L2 in the form of midline enhanced positivity within 300–500 ms. Then, within a later time window of 500–800 ms, the P600 component was observed for gender incongruity in both the L1 and the L2. However, large variance was observed in the participants' behavioral performance in the non-native language, and the amplitude of the P600 effect decreased linearly with the participants’ inability to detect gender incongruity. The above findings demonstrate that the rule of gender constraint functions similarly in L1 and L2 reflexive pronoun resolution by Chinese-English bilinguals, but nevertheless the exact mechanism employed for the L1 and the L2 was not the same, with an LAN-P600 pattern for the L1 but an Earlier Positivity-P600 pattern for the L2. Moreover, the large variance in L2 processing should be taken into consideration in future research on L2 reflexive pronoun resolution.

Fritz John and Karush–Kuhn–Tucker necessary conditions for local LU-optimal solutions of the constrained interval-valued optimization problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators are established. Under suitable assumptions on the generalized convexity of objective and constraint functions, sufficient conditions for LU-optimal solutions are given. The dual problems of Mond–Weir and Wolfe types are studied together with weak and strong duality theorems for them.

This paper is concerned with an uncertain multiobjective optimization problem, where the objective and constraint functions are sums-of-squares--convex (SOS-convex) polynomials, and the uncertainty...

We propose an inexact proximal bundle method for constrained nonsmooth nonconvex optimization problems whose objective and constraint functions are known through oracles which provide inexact information. The errors in function and subgradient evaluations might be unknown, but are merely bounded. To handle the nonconvexity, we first use the redistributed idea, and consider even more difficulties by introducing inexactness in the available information. We further examine the modified improvement function for a series of difficulties caused by the constrained functions. The numerical results show the good performance of our inexact method for a large class of nonconvex optimization problems. The approach is also assessed on semi-infinite programming problems, and some encouraging numerical experiences are provided.

This paper presents a mixed-integer quadratically constrained programming (MIQCP) formulation for B-spline constraints. The formulation can be used to obtain an exact MIQCP reformulation of any spline-constrained optimization problem problem, provided that the polynomial spline functions are continuous. This reformulation allows practitioners to use a general-purpose MIQCP solver, instead of a special-purpose spline solver, when solving B-spline constrained problems. B-splines are a powerful and widely used modeling tool, previously restricted from optimization due to lack of solver support. This contribution may encourage practitioners to use B-splines to model constraint functions. However, as the numerical study suggests, there is still a large gap between the solve times of the general-purpose solvers using the proposed formulation, and the special-purpose spline solver CENSO, the latter being significantly lower.