Abstract We introduce a new sequential algorithm for the Standard Quadratic Programming Problem (StQP), which exploits a formulation of StQP as a Linear Program with Linear Complementarity Constraints (LPLCC). The algorithm is finite and guarantees at least in theory a $$\delta $$ δ -approximate global minimum for an arbitrary small $$\delta $$ δ , which is a global minimum in practice. The sequential algorithm has two phases. In Phase 1, Stationary Points (SP) with strictly decreasing objective function values are computed. Phase 2 is designed for giving a certificate of global optimality for the last SP computed in Phase 1. Two different Nonlinear Programming Formulations for LPLCC are proposed for each one of these phases, which are solved by efficient enumerative algorithms. New procedures for computing a lower bound for StQP are also proposed, which are easy to implement and give tight bounds in general. Computational experiments with a number of test problems from known sources indicate that the two-phase sequential algorithm is, in general, efficient in practice. Furthermore, the algorithm seems to be an efficient way to study the copositivity of a matrix by exploiting an StQP with this matrix.

Multiclass classification is an important and ongoing research subject in machine learning. Current support vector methods for multiclass classification implicitly assume that the parameters in the optimization problems are known exactly. However, in practice, the parameters have perturbations since they are estimated from the training data, which are usually subject to measurement noise. In this article, we propose linear and nonlinear robust formulations for multiclass classification based on the M-SVM method. The preliminary numerical experiments confirm the robustness of the proposed method.

Evolutionary programming (EP) is one of the main classes of evolutionary algorithms (EAs). Improving existing EAs is necessary in order to achieve better results and overcome their costly computational complexity. In this paper, we present a new version of EP called Directed Evolutionary Programming (DEP) in which more directing strategies with learned termination criteria are invoked to overcome some drawbacks of EP. In DEP, the mutated children are given the chance to improve themselves with the guidance of their parents. The search process in DEP is supported by diversification and intensification schemes in order to keep the diversity, achieve faster convergence and equip the search with an automatic termination criteria. The computational experiments show that DEP is efficient and cheaper than some well-known versions of EP.

Abstract We study the multicommodity flow problem that minimizes a strictly convex cost objective function subject to the capacity constraints on individual commodities as well as the total flows in each arc. By making use of its dual, we formulate the problem as a nonlinear unconstrained optimization problem and propose relaxation methods. Computational results show that the proposed methods can practically solve problem instances, for example, with up to 100 nodes, 1000 arcs, and seven commodities.

A new method of ranking based on an indicator estimating the relative position of an element determined from the preference relation on the elements to be classified is proposed. The method is then applied to the multiple criteria decision problem and is illustrated by a simple hypothetical example. Possible developments and extensions of the method to various decision environments are explored.