Journal of Global Optimization最新文献

We are interested in assessing the use of neural networks as surrogate models to approximate and minimize objective functions in optimization problems. While neural networks are widely used for machine learning tasks such as classification and regression, their application in solving optimization problems has been limited. Our study begins by determining the best activation function for approximating the objective functions of popular nonlinear optimization test problems, and the evidence provided shows that ReLU and SiLU exhibit the best performance on both training and testing data. We then analyze the accuracy of function value, gradient, and Hessian approximations for such objective functions obtained through interpolation/regression models and neural networks. When compared to interpolation/regression models, neural networks can deliver competitive zero- and first-order approximations (at a high training cost) but underperform on second-order approximation. However, it is shown that combining a neural net activation function with the natural basis for quadratic interpolation/regression can waive the necessity of including cross terms in the natural basis, leading to models with fewer parameters to determine. Lastly, we provide evidence that the performance of a state-of-the-art derivative-free optimization algorithm can hardly be improved when the gradient of an objective function is approximated using any of the surrogate models considered, including neural networks.

In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios.

Bilevel optimization problems embed the optimality of a subproblem as a constraint of another optimization problem. We introduce the concept of near-optimality robustness for bilevel optimization, protecting the upper-level solution feasibility from limited deviations from the optimal solution at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of general bilevel optimization problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of exact and heuristic methods and the impact of valid inequalities on the solution time.

In this paper, we provide a necessary and sufficient condition under which the compositions of finitely many firmly nonexpansive mappings in a p-uniformly convex space converge strongly. In particular, we obtain convergence results for the method of alternating projections in CAT((kappa )) spaces, with (kappa ge 0). We also study the rate of convergence of the sequence generated by the method in cases where the firmly nonexpansive mappings satisfy an error bound, and the set of fixed points of these mappings is boundedly linearly regular.

In this paper, we address the vertex-traversing-constrained mixed Chinese postman problem (the VtcMCP problem), which is a further generalization of the Chinese postman problem, and this new problem has many practical applications in real life. Specifically, given a connected mixed graph (G=(V, Ecup A; w,b)) with length function (w(cdot )) on edges and arcs and traversal function (b(cdot )) on vertices, we are asked to determine a tour traversing each link (i.e., either edge or arc) at least once and each vertex v at most b(v) times, the objective is to minimize the total length of such a tour, where (n=|V|) is the number of vertices and (m=|Ecup A|) is the number of links of G, respectively. We obtain the following four main results. (1) Given any two constants (beta ge 1) and (alpha ge 1), we prove that there is no polynomial-time algorithm with approximation ratios ((1,beta )) or ((alpha , 1)) for solving the VtcMCP problem, where an (h, k)-approximation algorithm for solving the VtcMCP problem is one algorithm that produces a solution with violating the vertex-traversing constraints by at most a ratio of h and with costing at most k times the optimal value; (2) We design a (3, 2)-approximation algorithm ({{mathcal {A}}}) to solve the VtcMCP problem in time (O(m^{2}log n)); (3) We prove the fact that this algorithm ({{mathcal {A}}}) in (2) is indeed an exact algorithm to optimally solve the VtcMCP problem for the case (E=emptyset ); (4) We present an exact algorithm to optimally solve the VtcMCP problem in time (O(m^{3})) for the case (A=emptyset ).

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

We propose a new parametric approach to convex multiplicative programming problem. This problem is nonconvex optimization problem with a lot of practical applications. Compared with preceding methods based on branch-and-bound procedure and other approaches, the idea of our method is to reduce the original nonconvex problem to a parametric convex programming problem having parameters in objective functions. To find parameters corresponding to the optimal solution of the original problem, a system of nonlinear equations which the parameters should satisfy is studied. Then, the system is solved by a Newton-like algorithm, which needs to solve a convex programming problem in each iteration and has global linear and local superlinear/quadratic rate of convergence under some assumptions. Moreover, under some mild assumptions, our algorithm has a finite convergence, that is, the algorithm finds a solution after a finite number of iterations. The numerical results show that our method has much better performance than other reported methods for this class of problems.

We present a proximal point type algorithm tailored for tackling pseudomonotone equilibrium problems in a Hilbert space which are not necessarily convex in the second argument of the involved bifunction. Motivated by the extragradient algorithm, we propose a two-step method and we prove that the generated sequence converges strongly to a solution of the nonconvex equilibrium problem under mild assumptions and, also, we establish a linear convergent rate for the iterates. Furthermore, we identify a new class of functions that meet our assumptions, and we provide sufficient conditions for quadratic fractional functions to exhibit strong quasiconvexity. Finally, we perform numerical experiments comparing our algorithm against two alternative methods for classes of nonconvex mixed variational inequalities.

Based on the equivalent optimization problems of the splitting feasibility problem, we investigate this problem by using modified general splitting method in this paper. One is a relaxation splitting method with linearization, and the other combines the former with alternated inertial extrapolation step. The strong convergence of our algorithms is analyzed when related parameters are properly chosen. Compared with most existing results where inertial factor must be less than 1, inertial factor can be taken 1 in our alternated inertial-type algorithm. The efficiency of our methods are illustrated by some numerical examples.

The theory of duality is of fundamental importance in the study of vector optimization problems and vector equilibrium problems. A Mond–Weir-type dual model for such problems is important in practice. Therefore, studying such problems with a dual approach is really useful and necessary in the literature. The goal of this article is to formulate Mond–Weir-type dual models for the minimization problem (P), the constrained vector optimization problem (CVOP) and the constrained vector equilibrium problem (CVEP) in terms of (epsilon )-upper convexificators. By applying the concept of (epsilon )-pseudoconvexity, some weak, strong and converse duality theorems for the primal problem (P) and its dual problem (DP), the primal vector optimization problem (CVOP) and its Mond–Weir-type dual problem (MWCVOP), the primal vector equilibrium problem (P) and its Mond–Weir-type dual problem (MWCVEP) are explored.