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Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples. [ABSTRACT FROM AUTHOR]

This study addresses the scheduling problem of two identical parallel machines with the objective of minimizing the total completion time under the machine availability constraints. To the best of our knowledge, this study is the first to develop a fully polynomial-time approximation scheme (FPTAS), a solution method which has been neglected in past studies, to solve the studied problem. The FPTAS, which is based on a dynamic programming algorithm is developed by applying a trimming-the-state-space approach. Theoretical proofs of the error bound and the time complexity for the proposed FPTAS are also provided. The computational results indicate that the proposed FPTAS performs more efficiently than a dynamic programming algorithm in terms of both run time and problem size. The error bound of the FPTAS is demonstrated to be within the pre-specified error bound. [ABSTRACT FROM AUTHOR]

In this article, we aim to solve Bayesian Risk Optimization (BRO), which is a recently proposed framework that formulates simulation optimization under input uncertainty. In order to efficiently solve the BRO problem, we derive nested stochastic gradient estimators and propose corresponding stochastic approximation algorithms. We show that our gradient estimators are asymptotically unbiased and consistent, and that the algorithms converge asymptotically. We demonstrate the empirical performance of the algorithms on a two-sided market model. Our estimators are of independent interest in extending the literature of stochastic gradient estimation to the case of nested risk measures. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonsmooth vector optimization problem involving locally Lipschitz generalized approximate convex functions and find some relations between approximate convexity and generalized approximate convexity. We establish relationships between vector variational inequalities and nonsmooth vector optimization problem using the generalized approximate convexity as a tool. [ABSTRACT FROM AUTHOR]

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this work, we combine outer-approximation (OA) and bundle method algorithms for dealingwithmixed-integer non-linear programming (MINLP) problems with nonsmooth convex objective and constraint functions. As the convergence analysis of OA methods relies strongly on the differentiability of the involved functions, OA algorithms may fail to solve general nonsmooth convex MINLP problems. In order to obtain OA algorithms that are convergent regardless the structure of the convex functions, we solve the underlying OA's non-linear subproblems by a specialized bundle method that provides necessary information to cut off previously visited (non-optimal) integer points. This property is crucial for proving (finite) convergence of OA algorithms. We illustrate the numerical performance of the given proposal on a class of hybrid robust and chanceconstrained problems that involve a random variable with finite support. [ABSTRACT FROM AUTHOR]

We deal with a recently proposed method of Chubanov [A polynomial projection algorithm for linear feasibility problems. Math. Program. 153 (2015), pp. 687–713] for solving linear homogeneous systems with positive variables. Some improvements of Chubanov's method and its analysis are presented. We propose a new and simple cut criterion and show that the cuts defined by the new criterion are at least as sharp as in [1]. The new cut criterion reduces the iteration bound for his Basic Procedure by a factor 5, without changing the order of its strongly polynomial complexity. Our Modified Main Algorithm is in essence the same as Chubanov's Main Algorithm, except that it uses our Modified Basic Procedure as a subroutine. It is shown that it hastime complexity, just as in [1]. Some promising computational results are presented, in comparison with the optimization package Gurobi. [ABSTRACT FROM AUTHOR]

We describe general heuristics to approximately solve a wide variety of problems with convex objective and decision variables from a non-convex set. The heuristics, which employ convex relaxations, convex restrictions, local neighbour search methods, and the alternating direction method of multipliers, require the solution of a modest number of convex problems, and are meant to apply to general problems, without much tuning. We describe an implementation of these methods in a package called NCVX, as an extension of CVXPY, a Python package for formulating and solving convex optimization problems. We study several examples of well known non-convex problems, and show that our general purpose heuristics are effective in finding approximate solutions to a wide variety of problems. [ABSTRACT FROM AUTHOR]

A lot of heuristic algorithms, such as Evolutionary algorithms (EAs), are used to solve the maximum leaf spanning tree (MLST) problem which is non-deterministic polynomial time hard (NP-hard). However, the performance analysis of EAs on the MLST problem has seldom been studied theoretically. In this paper, we theoretically analyze the performance of the (1+1) EA on the MLST problem. We demonstrate that the (1+1) EA obtains-approximation ratio and 3-approximation ratio on this problem in expected polynomial runtimeO(nm2) andO(nm4), respectively, wherenis the number of nodes andmis the number of edges in a connected undirected graph. Furthermore, we reveal that the (1+1) EA can outperform the local search algorithms on two instances of the MLST problem. [ABSTRACT FROM AUTHOR]

We survey approximation issues matching ideas and tools from the fields of polynomial approximation, moderately exponential computation and fixed parameter tractability aiming at designing approximation algorithms achieving ratios unachievable in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. This research programme, despite its relative youth, is becoming very active in computer science and in combinatorial optimization and motivates extensive research. [ABSTRACT FROM PUBLISHER]

In this article, the job shop scheduling problem with two batch-processing machines is considered. The machines have limited capacity and the jobs have non-identical job sizes. The jobs are processed in batches and the total size of each batch cannot exceed the machine capacity. The processing times of a job on the two machines are proportional. We show the problem of minimising makespan is NP-hard in the strong sense. Then we provide an approximation algorithm with worst-case ratio no more than 4, and the running time of the algorithm is O(n log n). Finally, the performance of the proposed algorithm is tested by different levels of instances. Computational results demonstrate the effectiveness of the algorithm for all the instances. [ABSTRACT FROM PUBLISHER]