Journal of Global Optimization最新文献

We consider solving convex problems satisfying quadratic growth condition (QGC) over a distributed setting with no central server. Such problems are popular in distributed machine learning applications. When QGC growth parameter c is known, we propose distributed accelerated gradient methods with restarts, named PDACA and DACA respectively for constrained and unconstrained settings. In practical problems when c is unavailable, we design mPDACA and mDACA methods respectively for constrained and unconstrained settings, where novel distributed mechanisms are proposed to update the estimates of growth parameter c using only local quantities depending on local proximal operators or local gradients. We further derive theoretical guarantees and gradient computation and communication complexities for all four proposed algorithms. Extensive numerical experiments on logistic regression on different communication topologies showcase the utility of our algorithms in comparison with baseline methods.

We study relaxations for linear programs with complementarity constraints, especially instances whose complementary pairs of variables are not independent. Our formulation is based on identifying vertex covers of the conflict graph of the instance and contains the extended formulation obtained from the ERLT introduced by Nguyen, Richard, and Tawarmalani as a special case. We demonstrate how to obtain strong cutting planes for our formulation from both the stable set polytope and the boolean quadric polytope associated with a complete bipartite graph. Through an extensive computational study for three types of practical problems, we assess the performance of our proposed linear relaxation and new cutting-planes in terms of the optimality gap closed.

Focusing on the extreme points of the solution sets of matrix two-person games, we propose the notions of characteristic sets and characteristic numbers. The characteristic sets (resp., the characteristic numbers) are the sets (resp., the numbers) of the extreme points of the solution set of the game and the optimal solution sets of the players. These concepts allow us to measure the complexity of the game. The larger the characteristic numbers, the more complex the game is. Among other things, we obtain upper bounds for the characteristic numbers and give a novel geometric construction. By the construction, we get useful descriptions of the optimal strategy set of each player of a game given by any nonsingular square matrix. Namely, the investigation of the geometry the just-mentioned sets reduces to computing or studying certain simpler sets. We also formulate several open problems.

In this paper, we further study the projected-type method for the extended vertical linear complementarity problem. By making use of some basic absolute value inequalities, some new convergence properties of the projected-type method are obtained. Compared with the existing results in the literature, the convergence range of the projected-type method is enlarged. By several numerical experiments, we also show the performance of the projected-type method.

By using a constant step-size, the convergence analysis of the gradient projection method on the sphere is presented for a closed spherically convex set. This algorithm is applied to discuss copositivity of operators with respect to cones. This approach can also be used to analyse solvability of nonlinear cone-complementarity problems. To our best knowledge this is the first numerical method related to the copositivity of operators with respect to the positive semidefinite cone. Numerical results concerning the copositivity of operators are also provided.

We introduce a new optimization approach to solving systems of split equality problems in real Hilbert spaces. We use the inertial method in order to improve the convergence rate of the proposed algorithms. Our algorithms do not depend on the norms of the bounded linear operators which appear in each split equality problem of the system under consideration. This is also a strong point of our algorithms because it is known that it is difficult to compute or estimate the norm of a linear operator in the general case.

The role of an expert in the decision-making process is crucial. If we ask an expert to help us to make a decision we assume their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is for an expert to provide manipulated data in a given case of decision-making process becomes essential. In the presented work, we consider manipulation of a ranking obtained by the Geometric Mean Method applied to a pairwise comparisons matrix. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives in a ranking. We also define a new index which measures how difficult such manipulation is in a given case.

In this paper, we perform optimality conditions and sensitivity analysis for parametric nonconvex minimax programming problems. Our aim is to study the necessary optimality conditions by using the Mordukhovich (limiting) subdifferential and to give upper estimations for the Mordukhovich subdifferential of the optimal value function in the problem under consideration. The optimality conditions and sensitivity analysis are obtained by using upper estimates for Mordukhovich subdifferentials of the maximum function. The results on optimality conditions are then applied to parametric multiobjective optimization problems. An example is given to illustrate our results.

Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an (varepsilon )-stationary point is upper bounded by ({mathcal {O}}(varepsilon ^{-3})). Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm.

In this paper, a partial Bregman alternating direction method of multipliers (ADMM) with a general relaxation factor (alpha in (0,frac{1+sqrt{5}}{2})) is proposed for structured nonconvex and nonsmooth optimization, where the objective function is the sum of a nonsmooth convex function and a smooth nonconvex function without coupled variables. We add a Bregman distance to alleviate the difficulty of solving the nonsmooth subproblem. For the smooth subproblem, we directly perform a gradient descent step of the augmented Lagrangian function, which makes the computational cost of each iteration of our method very cheap. To our knowledge, the nonconvex ADMM with a relaxation factor (alpha ne 1) in the literature has never been studied for the problem under consideration. Under some mild conditions, the boundedness of the generated sequence, the global convergence and the iteration complexity are established. The numerical results verify the efficiency and robustness of the proposed method.