Nonsmooth Multiobjective Optimization Research Articles

On optimality conditions and duality for multiobjective fractional optimization problem with vanishing constraints

<p>The aim of this paper is to investigate the optimality conditions for a class of nonsmooth multiobjective fractional optimization problems subject to vanishing constraints. In particular, necessary and sufficient conditions for (weak) Pareto solution are presented in terms of the Clark subdifferential. Furthermore, we construct Wolfe and Mond–Weir-type dual models and derive some duality theorems by using generalized quasiconvexity assumptions. Some examples to show the validity of our conclusions are provided.</p>

A Hybrid Chaotic-Based Multiobjective Differential Evolution Technique for Economic Emission Dispatch Problem

The economic emission dispatch problem (EEDP) is a nonconvex and nonsmooth multiobjective optimization problem in the power system field. Generally, fuel cost and total emissions of harmful gases are the problem objective functions. The EEDP decision variables are output powers of thermal generating units (TGUs). To make the EEDP problem more practical, valve point loading effects (VPLEs), prohibited operation zones (POZs), and power balance constraints should be included in the problem constraints. In order to solve this complex and constrained EEDP, a new multiobjective optimization technique combining the differential evolution (DE) algorithm and chaos theory is proposed in this study. In this new multiobjective optimization technique, a nondomination sorting principle and a crowding distance calculation are employed to extract an accurate Pareto front. To avoid being trapped in local optima and enhance the conventional DE algorithm, two different chaotic maps are used in its initialization, crossover, and mutation phases instead of random numbers. To overcome difficulties caused by the equality constraint describing the power balance constraint, a slack TGU is defined to compensate for the gap between the total generation and the sum of the system load and total power losses. Then, the optimal power outputs of all thermal units except the slack unit are determined by the suggested optimization technique. To assess the effectiveness and applicability of the proposed method for solving the EEDP, the six-unit and ten-unit systems are used. Moreover, obtained results are compared with other new optimization techniques already developed and tested for the same purpose. The superior performance of the ChMODE is also evaluated by using various metrics such as inverted generational distance (IGD), hyper-volume (HV), spacing metric (SM), and the average satisfactory degree (ASD).

Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints

This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of the considered problem is derived. The concept of ε proper Abadie data qualification is also introduced, and a necessary condition which is called a strictly strong stationary condition for Borwein proper efficient solutions is obtained. In view of the strictly strong stationary condition, convexity of the objective functions, and quasi-convexity of constrained functions, sufficient conditions for the Borwein proper efficient solutions are presented. Some examples are given to illustrate the reasonability of the obtained results.

Quasi Efficient Solutions and Duality Results in a Multiobjective Optimization Problem with Mixed Constraints via Tangential Subdifferentials

We take up a nonsmooth multiobjective optimization problem with tangentially convex objective and constraint functions. In employing a suitable constraint qualification, we formulate both necessary and sufficient optimality conditions for (local) quasi efficient solutions in terms of tangential subdifferentials. Furthermore, under generalized convexity assumptions, we state strong, weak and converse duality relations of Wolfe and Mond–Weir types. We give a number of examples to illustrate the new concepts and main results of this paper.

Necessary optimality conditions for nonsmooth multi-objective bilevel optimization problem under the optimistic perspective

Necessary optimality conditions for nonsmooth multi-objective bilevel optimization problem under the optimistic perspective

On Minty variational principle for nonsmooth multiobjective optimization problems on Hadamard manifolds

In this paper, we consider classes of approximate Minty and Stampacchia type vector variational inequalities using Clarke subdifferential on Hadamard manifolds and a class of nonsmooth multiobjective optimization problems. We investigate the relationship between the solution of these approximate vector variational inequalities and the solution of nonsmooth multiobjective optimization problems involving geodesic approximately convex functions. The results presented in this paper extend and generalize some existing results in the literature.

A note on characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators

This note concerns with Theorem 4.1 in [Kabgani A, Soleimani-damaneh M. Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization, 2018; 67: 217–235]. We present this theorem in a correct way and show that the error does not influence other results of the paper. Moreover, we provide some new corollaries for this theorem.

Second-order optimality conditions and duality for nonsmooth multiobjective optimization problems

Second-order optimality conditions and duality for nonsmooth multiobjective optimization problems

On stationarity for nonsmooth multiobjective problems with vanishing constraints

The aim of this paper is to develop first-order necessary and sufficient optimality conditions for nonsmooth multiobjective optimization problems with vanishing constraints. First of all, we introduce some data qualifications for the problem, and derive the comparisons between them. Secondly, based on the mentioned qualifications, we demonstrate some necessary optimality conditions, named strongly stationary conditions, at weakly efficient and efficient solutions of considered problem. Then, we show that the strongly stationary conditions are also sufficient for optimality. Finally, using the tightened problems, we establish other classes of qualifications and stationary conditions.

Necessary optimality conditions in nonsmooth semi-infinite multiobjective optimization under metric subregularity

We consider nonsmooth semi-infinite multiobjective optimization problems under mixed constraints, including infinitely many mixed constraints by using Clarke subdifferential. Semi-infinite programming (SIP) is the minimization of many scalar objective functions subject to a possibly infinite system of inequality or/and equality constraints. SIPs have been proved to be very important in optimization and applications. Semi-infinite programming problems arise in various fields of engineering such as control systems design, decision making under competition, and multiobjective optimization. There is extensive literature on standard semi-infinite programming problems. The investigation of optimality conditions for these problems is always one of the most attractive topics and has been studied extensively in the literature. In our work, we study optimality conditions for weak efficiency of a multiobjective semi-infinite optimization problem under mixed constraints including infinitely many of both equality and inequality constraints in terms of Clarke subdifferential. Our conditions are the form of the Karush-Kuhn-Tucker (KKT) multiplier. To the best of our knowledge, only a few papers are dealing with optimality conditions for SIPs subject to mixed constraints. By the Pshenichnyi-Levin-Valadire (PLV) property and the directional metric subregularity, we introduce a type of Mangasarian-Fromovitz constraint qualification (MFCQ). Then we show that (MFCQ) is a sufficient condition to guarantee the extended Abadie constraint qualification (ACQ) to satisfy. In our constraint qualifications, all functions are nonsmooth and the number of constraints is not necessarily finite. In our paper, we do not need the involved functions: convexity and differentiability. Later, we apply the extended Abadie constraint qualification to get the KKT multipliers for weak efficient solutions of SIP. Many examples are provided to illustrate some advantages of our results. The paper is organized as follows. In Section Preliminaries, we present our basic definitions of nonsmooth and convex analysis. Section Main Results prove necessary conditions for the weakly efficient solution in terms of the Karush-Kuhn-Tucker mult iplier rule with the help of some constraint qualifications.

Characterizations of robust ε-quasi optimal solutions for nonsmooth optimization problems with uncertain data

ABSTRACT This paper deals with robust ε-quasi optimal solutions for a class of nonsmooth optimization problems with uncertain data. Under some mild assumptions, we first establish, by using robust optimization (i.e. worst-case) approach, approximate optimality conditions for this uncertain nonsmooth optimization problem. Then, we introduce a Mixed-type robust approximate dual problem of this uncertain optimization problem, and explore their relationships. Moreover, using a scalarization method, we derive optimality conditions for robust weakly approximate efficient solutions for an uncertain nonsmooth multiobjective optimization problem. We also obtain approximate duality theorems for the uncertain nonsmooth multiobjective optimization problem.

An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems

We present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method with the multiobjective proximal bundle method by Mäkelä. The results indicate that our method is competitive while being easier to implement. Although the number of objective function evaluations is larger, the overall number of subgradient evaluations is smaller. Our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth problems. Finally, we demonstrate how our method can be used for solving sparse optimization problems, which are present in many real-life applications.

Robust Optimality and Duality in Multiobjective Optimization Problems under Data Uncertainty

In this paper, we employ advanced techniques of variational analysis and generalized differentiation to examine robust optimality conditions and robust duality for an uncertain nonsmooth multiobjective optimization problem under arbitrary uncertainty nonempty sets. We establish necessary and sufficient optimality conditions for (local) robust (weakly) efficient solutions of the considered problem. Our problem involves nonsmooth real-valued functions and data uncertainty in both the objective and constraint functions, and its necessary and sufficient optimality conditions are exhibited in terms of multipliers and the Mordukhovich or Clarke subdifferentials of the related functions. Moreover, we formulate a dual multiobjective problem to the underlying program and examine robust weak, strong, and converse duality relations between the primal problem and its dual under assumptions of (strictly) generalized convexity.

Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization

Several countries utilize nuclear power and face the problem of what to do with the spent nuclear fuel. One possibility, which is under the scope in this paper, is to dispose of the fuel assemblies in the disposal facility. Before the assemblies can be disposed of, they must cool down their decay heat power in the interim storage. Next, they are loaded into canisters in the encapsulation facility, and finally, the canisters are placed in the disposal facility. In this paper, we model this process as a nonsmooth multiobjective mixed-integer nonlinear optimization problem with the minimization of nine objectives: the maximum number of assemblies in the storage, maximum storage time, average storage time, total number of canisters, end time of the encapsulation, operation time of the encapsulation facility, the lengths of disposal and central tunnels, and total costs. As a result, we obtain the disposal schedule i.e., amount of canisters disposed of periodically. We introduce the interactive multiobjective optimization method using the two-slope parameterized achievement scalarizing functions which enables us to obtain systematically several different Pareto optimal solutions from the same preference information. Finally, a case study adapting the disposal in Finland is given. The results obtained are analyzed in terms of the objective values and disposal schedules.

Necessary conditions for weak efficiency for nonsmooth degenerate multiobjective optimization problems

In this paper we give primal first and second-order necessary conditions for the existence of a local weak minimum for nonsmooth multiobjective optimization problems with inequality constraints and an arbitrary constraint set. For nonsmooth multiobjective problems with inequality and degenerate equality constraints, we present primal necessary conditions and Kuhn–Tucker type dual necessary conditions under a new constraint qualification. The effectiveness of our results is illustrated on some examples.

Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly introduce two kinds of generalized convex functions, which are not necessary to be convex. Robust necessary optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are established by a generalized alternative theorem and the robust constraint qualification. Further, robust sufficient optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are also derived. The Mond–Weir-type dual problem and Wolfe-type dual problem are formulated. Finally, we obtain the weak, strong and converse robust duality results between the primal one and its dual problems under the generalized convexity assumptions.

Characterization of Isolated Efficient Solutions in Nonsmooth Multiobjective Semi-infinite Programming

The purpose of this paper is to characterize the isolated efficient solutions of a given nonsmooth multiobjective optimization problem with finitely many objective functions and infinitely many constraints. To do this, we introduce a new class of functions in order to get Karush–Kuhn–Tucker type necessary and sufficient optimality conditions for the problem. This article can be considered as a supplement to the recent paper by Kanzi (Iran J Sci Technol Trans Sci 42:1537–1544, 2018).

Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds

A proximal point method for nonsmooth multiobjective optimization in the Riemannian context is proposed, and an optimality condition for multiobjective problems is introduced. This allowed replacing the classic approach, via “scalarization,” by a purely vectorial and considering the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Moreover, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto optimal is obtained.

First-order necessary conditions in locally Lipschitz multiobjective optimization

In this paper, some nonsmooth analogues of the Burachik–Rizvi regularity conditions are introduced for a nonsmooth multiobjective optimization problem with equality and inequality constraints and with an arbitrary set constraint. Primal first-order necessary conditions for local efficiency and local Geoffrion-proper efficiency are given for this problem. Dual Kuhn–Tucker necessary conditions for local efficiency and local Geoffrion-proper efficiency are obtained for the multiobjective problem with equality and inequality constraints.

Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.