Journal of Global Optimization最新文献

Let P be a set of n points in (mathbb {R}^3) in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal (O(nlog n)) time and O(n) space algorithm to compute RCH(P). We also obtain an efficient (O(nlog ^2 n)) time and (O(nlog n)) space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate ({mathbb {R}}^3) around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in (mathbb {R}^3). Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in (mathbb {R}^3).

The aim of this paper is to investigate the problem of designing and building International Environmental Agreements (IEAs) taking into account some normative properties. We consider n asymmetric countries of the world, each one generating a quantity of pollutant emissions from the production of goods and services. We assume that individual emissions yield private benefits and negative externalities affecting all countries. To determine its own level of pollution, each state conducts a cost-benefit analysis. The absence of a supranational entity imposing emissions reduction makes IEAs based on voluntary participation. Examining the standard static non-cooperative game-theoretical model of coalition formation, we discover that the resulting emissions allocations might not be equitable à la Foley. It means that there might exist at least one player preferring to implement some other agent’s strategic plan instead of to play her own strategy. With the goal of studying whether equity, at least among coalesced countries, may be a criterion leading to social improvement, we introduce a new optimization rule. We require that members of an environmental coalition have to solve the maximization problem subject to the constraint imposing that they do not envy each other. Analyzing the particular case of two-player games, we get that, when countries are, in a sense, not too different from each other, our new mechanism endogenously induces social equity. By imposing a suitable total emission cap, the same results extend to all those games where our and standard solutions coexist and are different.

This paper investigates sparse optimization problems characterized by a sparse group structure, where element- and group-level sparsity are jointly taken into account. This particular optimization model has exhibited notable efficacy in tasks such as feature selection, parameter estimation, and the advancement of model interpretability. Central to our study is the scrutiny of the (ell _0) and (ell _{2,0}) norm regularization model, which, in comparison to alternative surrogate formulations, presents formidable computational challenges. We embark on our study by conducting the analysis of the optimality conditions of the sparse group optimization problem, leveraging the notion of a (gamma )-stationary point, whose linkage to local and global minimizer is established. In a subsequent facet of our study, we develop a novel subspace Newton algorithm for sparse group (ell _0) optimization problem and prove its global convergence property as well as local second-order convergence rate. Experimental results reveal the superlative performance of our algorithm in terms of both precision and computational expediency, thereby outperforming several state-of-the-art solvers.

Bi-quadratic programming over unit spheres is a fundamental problem in quantum mechanics introduced by pioneer work of Einstein, Schrödinger, and others. It has been shown to be NP-hard; so it must be solve by efficient heuristic algorithms such as the block improvement method (BIM). This paper focuses on the maximization of bi-quadratic forms with nonnegative coefficient tensors, which leads to a rank-one approximation problem that is equivalent to computing the M-spectral radius and its corresponding eigenvectors. Specifically, we propose a tight upper bound of the M-spectral radius for nonnegative fourth-order partially symmetric (PS) tensors. This bound, serving as an improved shift parameter, significantly enhances the convergence speed of BIM while maintaining computational complexity aligned with the initial shift parameter of BIM. Moreover, we elucidate that the computation cost of such upper bound can be further simplified for certain sets and delve into the nature of these sets. Building on the insights gained from the proposed bounds, we derive the exact solutions of the M-spectral radius and its corresponding M-eigenvectors for certain classes of fourth-order PS-tensors and discuss the nature of this specific category. Lastly, as a practical application, we introduce a testable sufficient condition for the strong ellipticity in the field of solid mechanics. Numerical experiments demonstrate the utility of the proposed results.

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.

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.

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.

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.