Optimization Letters最新文献

When faced with a limited budget of function evaluations, state-of-the-art black-box optimization (BBO) solvers struggle to obtain globally, or sometimes even locally, optimal solutions. In such cases, one may pursue solution polishing, i.e., a computational method to improve (or “polish”) an incumbent solution, typically via some sort of evolutionary algorithm involving two or more solutions. While solution polishing in “white-box” optimization has existed for years, relatively little has been published regarding its application in costly-to-evaluate BBO. To fill this void, we explore two novel methods for performing solution polishing along one-dimensional curves rather than along straight lines. We introduce a convex quadratic program that can generate promising curves through multiple elite solutions, i.e., via path relinking, or around a single elite solution. In comparing four solution polishing techniques for continuous BBO, we show that solution polishing along a curve is competitive with solution polishing using a state-of-the-art BBO solver.

This paper proposes and analyzes the notion of dual cones associated with the metric projection and generalized projection in Banach spaces. We show that the dual cones, related to the metric projection and generalized metric projection, lose many important properties in transitioning from Hilbert spaces to Banach spaces. We also propose and analyze the notions of faces and visions in Banach spaces and relate them to metric projection and generalized projection. We provide many illustrative examples to give insight into the given results

Current methodologies for finding portfolio rules under the Merton framework employ hard-to-implement numerical techniques. This work presents a methodology that can derive an allocation à la Merton in a spreadsheet, under an incomplete market with a time-varying dividend yield and long-only constraints. The first step of the method uses the martingale approach to obtain a portfolio rule in a complete artificial market. The second step derives a closed-form optimal solution satisfying the long-only constraints, from the unconstrained solution of the first step. This is done by determining closed-form Lagrangian dual processes satisfying the primal-dual optimality conditions between the true and artificial markets. The last step estimates the parameters defined in the artificial market, to then obtain analytical approximations for the hedging demand component within the optimal portfolio rule of the previous step. The methodology is tested with real market data from 16 US stocks from the Dow Jones. The results show that the proposed solution delivers higher financial wealth than the myopic solution, which does not consider the time-varying nature of the dividend yield. The sensitivity analysis carried out on the closed-form solution reveals that the difference with respect to the myopic solution increases when the price of the risky asset is more sensitive to the dividend yield, and when the dividend yield presents a higher probability of diverging from the current yield. The proposed solution also outperforms a known Merton-type solution that derives the Lagrangian dual processes in another way.

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) (Ax-|x|=b). The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever (rho (|A^{-1}|)<1/3). We also present new results for the case where (A-I) is a nonsingular M-matrix or an irreducible singular M-matrix. When (A-I) is an irreducible singular M-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.

We consider the problem of determining the optimal node interception strategy during influence propagation over a (directed) network (G=(V,A)). More specifically, this work aims to find an interception set (D subseteq V) such that the influence spread over the remaining network (G backslash D) under the linear threshold diffusion model is minimized. We prove its NP-hardness, even in the case when G is an undirected graph with unit edge weights. An exact algorithm based on integer linear programming and delayed constraint generation is proposed to determine the most critical nodes in the influence propagation process. Additionally, we investigate the technique of lifting inequalities of minimal activation sets. Experiments on the connected Watts-Strogatz small-world networks and real-world networks are also conducted to validate the effectiveness of our methodology.

In this article, we study weak stationarity conditions (A- and C-) for a particular class of degenerate stochastic mathematical programming problems with complementarity constraints (SMPCC, for short). Importance of the weak stationarity concepts in absence of SMPCC-LICQ are presented through toy problems in which the point of local or global minimizers are weak stationary points rather than satisfying other stronger stationarity conditions. Finally, a well known technique to solve stochastic programming problems, namely sample average approximation (SAA) method, is studied to show the significance of the weak stationarity conditions for degenerate SMPCC problems. Consistency of weak stationary estimators are established under weaker constraint qualifications than SMPCC-LICQ.

We propose a multistart algorithm to identify all local minima of a constrained, nonconvex stochastic optimization problem. The algorithm uniformly samples points in the domain and then starts a local stochastic optimization run from any point that is the “probabilistically best” point in its neighborhood. Under certain conditions, our algorithm is shown to asymptotically identify all local optima with high probability; this holds even though our algorithm is shown to almost surely start only finitely many local stochastic optimization runs. We demonstrate the performance of an implementation of our algorithm on nonconvex stochastic optimization problems, including identifying optimal variational parameters for the quantum approximate optimization algorithm.

In the game theoretical approach of the basic problem in Combinatorial Search an adversary thinks of a defective element d of an n-element pool X, and the questioner needs to find x by asking questions of type is (din Q?) for certain subsets Q of X. We study cooperative versions of this problem, where there are multiple questioners, but not all of them learn the answer to the queries. We consider various models that differ in how it is decided who gets to ask the next query, who obtains the answer to the query, and who needs to know the defective element by the end of the process.