Journal of Global Optimization最新文献

During the last decades many metaheuristics for global numerical optimization have been proposed. Among them, Basin Hopping is very simple and straightforward to implement, although rarely used outside its original Physical Chemistry community. In this work, our aim is to compare Basin Hopping, and two population variants of it, with readily available implementations of the well known metaheuristics Differential Evolution, Particle Swarm Optimization, and Covariance Matrix Adaptation Evolution Strategy. We perform numerical experiments using the IOH profiler environment with the BBOB test function set and two difficult real-world problems. The experiments were carried out in two different but complementary ways: by measuring the performance under a fixed budget of function evaluations and by considering a fixed target value. The general conclusion is that Basin Hopping and its newly introduced population variant are almost as good as Covariance Matrix Adaptation on the synthetic benchmark functions and better than it on the two hard cluster energy minimization problems. Thus, the proposed analyses show that Basin Hopping can be considered a good candidate for global numerical optimization problems along with the more established metaheuristics, especially if one wants to obtain quick and reliable results on an unknown problem.

We study the non-submodular maximization problem, in which the objective function is characterized by parameters, subject to a cardinality or (p)-system constraint. By adapting the Threshold-Greedy algorithm for the submodular maximization, we present two deterministic algorithms for approximately solving the non-submodular maximization problem. Our analysis shows that the algorithms we propose requires much less function evaluations than existing algorithms, while providing comparable approximation guarantees. Moreover, numerical experiment results are presented to validate the theoretical analysis. Our results not only fill a gap in the (non-)submodular maximization, but also generalize and improve several existing results on closely related optimization problems.

Abstract

This paper explores a class of nonlinear Adjustable Robust Optimization (ARO) problems, containing here-and-now and wait-and-see variables, with uncertainty in the objective function and constraints. By applying Fenchel’s duality on the wait-and-see variables, we obtain an equivalent dual reformulation, which is a nonlinear static robust optimization problem. Using the dual formulation, we provide conditions under which the ARO problem is convex on the here-and-now decision. Furthermore, since the dual formulation contains a non-concave maximization on the uncertain parameter, we use perspective relaxation and an alternating method to handle the non-concavity. By employing the perspective relaxation, we obtain an upper bound, which we show is the same as the static relaxation of the considered problem. Moreover, invoking the alternating method, we design a new dual-based cutting plane algorithm that is able to find a reasonable lower bound for the optimal objective value of the considered nonlinear ARO model. In addition to sketching and establishing the theoretical features of the algorithms, including convergence analysis, by numerical experiments we reveal the abilities of our cutting plane algorithm in producing locally robust solutions with an acceptable optimality gap.

In this paper, we investigate a generalized multiplicative problem (GMP) that is known to be NP-hard even with one linear product term. We first introduce some criterion-space variables to obtain an equivalent problem of the GMP. A criterion-space branch-reduction-bound algorithm is then designed, which integrates some basic operations such as the two-level linear relaxation technique, rectangle branching rule and criterion-space region reduction technologies. The global convergence of the presented algorithm is proved by means of the subsequent solutions of a series of linear relaxation problems, and its maximum number of iterations is estimated on the basis of exhaustiveness of branching rule. Finally, numerical results demonstrate the presented algorithm can efficiently find the global optimum solutions for some test instances with the robustness.

In this paper, we address the difficulty of solving large-scale multi-dimensional knapsack instances (MKP), presenting a novel deep reinforcement learning (DRL) framework. In this DRL framework, we train different agents compatible with a discrete action space for sequential decision-making while still satisfying any resource constraint of the MKP. This novel framework incorporates the decision variable values in the 2D DRL where the agent is responsible for assigning a value of 1 or 0 to each of the variables. To the best of our knowledge, this is the first DRL model of its kind in which a 2D environment is formulated, and an element of the DRL solution matrix represents an item of the MKP. Our framework is configured to solve MKP instances of different dimensions and distributions. We propose a K-means approach to obtain an initial feasible solution that is used to train the DRL agent. We train four different agents in our framework and present the results comparing each of them with the CPLEX commercial solver. The results show that our agents can learn and generalize over instances with different sizes and distributions. Our DRL framework shows that it can solve medium-sized instances at least 45 times faster in CPU solution time and at least 10 times faster for large instances, with a maximum solution gap of 0.28% compared to the performance of CPLEX. Furthermore, at least 95% of the items are predicted in line with the CPLEX solution. Computations with DRL also provide a better optimality gap with respect to state-of-the-art approaches.

An online majorized semi-proximal alternating direction method of multiplier (Online-mspADMM) is proposed for a broad class of online linearly constrained composite optimization problems. A majorized technique is adopted to produce subproblems which can be easily solved. Under mild assumptions, we establish (mathcal {O}(sqrt{N})) objective regret and (mathcal {O}(sqrt{N})) constraint violation regret at round N. We apply the Online-mspADMM to solve different types of online regularized logistic regression problems. The numerical results on synthetic data sets verify the theoretical result about regrets.

We propose a multi-swarm approach to approximate the Pareto front of general multi-objective optimization problems that is based on the consensus-based optimization method (CBO). The algorithm is motivated step by step beginning with a simple extension of CBO based on fixed scalarization weights. To overcome the issue of choosing the weights we propose an adaptive weight strategy in the second modeling step. The modeling process is concluded with the incorporation of a penalty strategy that avoids clusters along the Pareto front and a diffusion term that prevents collapsing swarms. Altogether the proposed K-swarm CBO algorithm is tailored for a diverse approximation of the Pareto front and, simultaneously, the efficient set of general non-convex multi-objective problems. The feasibility of the approach is justified by analytic results, including convergence proofs, and a performance comparison to the well-known non-dominated sorting genetic algorithms NSGA2 and NSGA3 as well as the recently proposed one-swarm approach for multi-objective problems involving consensus-based optimization.

Abstract

The problem of finding a globally optimal k-partition of a set  (mathcal {A}) is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one feature ( (mathcal {A}subset mathbb {R}) ), there is no method to solve. Only in the one-dimensional case, there are efficient methods based on the fact that the search for a globally optimal k-partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm DIRECT. In the present paper, we propose a method for finding a globally optimal k-partition in the general case ( (mathcal {A}subset mathbb {R}^n) , (nge 1) ), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the k-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the k-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient.

Interval branch-and-bound solvers provide reliable algorithms for handling non-convex optimization problems by ensuring the feasibility and the optimality of the computed solutions, i.e. independently from the floating-point rounding errors. Moreover, these solvers deal with a wide variety of mathematical operators. However, these solvers are not dedicated to quadratic optimization and do not exploit nonlinear convex relaxations in their framework. We present an interval branch-and-bound method that can efficiently solve quadratic optimization problems. At each node explored by the algorithm, our solver uses a quadratic convex relaxation which is as strong as a semi-definite programming relaxation, and a variable selection strategy dedicated to quadratic problems. The interval features can then propagate efficiently this information for contracting all variable domains. We also propose to make our algorithm rigorous by certifying firstly the convexity of the objective function of our relaxation, and secondly the validity of the lower bound calculated at each node. In the non-rigorous case, our experiments show significant speedups on general integer quadratic instances, and when reliability is required, our first results show that we are able to handle medium-sized instances in a reasonable running time.

This paper is devoted to the study of a class of multiobjective semi-infinite programming problems on Hadamard manifolds (in short, (MOSIP-HM)). We derive some alternative theorems analogous to Tucker’s theorem, Tucker’s first and second existence theorem, and Motzkin’s theorem of alternative in the framework of Hadamard manifolds. We employ Motzkin’s theorem of alternative to establish necessary and sufficient conditions that characterize KKT pseudoconvex functions using strong KKT vector critical points and efficient solutions of (MOSIP-HM). Moreover, we formulate the Mond-Weir and Wolfe-type dual problems related to (MOSIP-HM) and derive the weak and converse duality theorems relating (MOSIP-HM) and the dual problems. Several non-trivial numerical examples are provided to illustrate the significance of the derived results. The results deduced in the paper extend and generalize several notable works existing in the literature.