Outer Approximation Algorithms for Canonical DC Problems

The paper discusses a general framework for outer approximation type algorithms for the canonical DC optimization problem. The algorithms rely on a polar reformulation of the problem and exploit an approximated oracle in order to check global optimality. Consequently, approximate optimality conditions are introduced and bounds on the quality of the approximate global optimal solution are obtained. A thorough analysis of properties which guarantee convergence is carried out; two families of conditions are introduced which lead to design six implementable algorithms, whose convergence can be proved within a unified framework

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented

Resolution Method for Mixed Integer Linear Multiplicative-Linear Bilevel Problems Based on Decomposition Technique

I In this paper, we propose an algorithm base on decomposition technique for solving the mixed integer linear multiplicative-linear bilevel problems. In fact, this algorithm is an application of the algorithm given by G. K. Saharidis et al for the case in which the first level objective function is linear multiplicative. We use properties of quasi-concave of bilevel programming problems and decompose the initial problem into two subproblems named RM P and SP . The lower and upper bound provided from the RM P and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is less than an arbitrary tolerance. In conclusion, some numerical examples are presented in order to show the efficiency of algorithm

Solution to the generalized lattice point and related problems to disjunctive programming

Issued as Pre-prints [1-5], Progress reports [1-2], Final summary report, and Final technical report, Project no. E-24-67

Outer Approximation Algorithms for DC Programs and Beyond

We consider the well-known Canonical DC (CDC) optimization problem, relying on an alternative equivalent formulation based on a polar characterization of the constraint, and a novel generalization of this problem, which we name Single Reverse Polar problem (SRP). We study the theoretical properties of the new class of (SRP) problems, and contrast them with those of (CDC)problems. We introduce of the concept of ``approximate oracle'' for the optimality conditions of (CDC) and (SRP), and make a thorough study of the impact of approximations in the optimality conditions onto the quality of the approximate optimal solutions, that is the feasible solutions which satisfy them. Afterwards, we develop very general hierarchies of convergence conditions, similar but not identical for (CDC) and (SRP), starting from very abstract ones and moving towards more readily implementable ones. Six and three different sets of conditions are proposed for (CDC) and (SRP), respectively. As a result, we propose very general algorithmic schemes, based on approximate oracles and the developed hierarchies, giving rise to many different implementable algorithms, which can be proven to generate an approximate optimal value in a finite number of steps, where the error can be managed and controlled. Among them, six different implementable algorithms for (CDC) problems, four of which are new and can't be reduced to the original cutting plane algorithm for (CDC) and its modifications; the connections of our results with the existing algorithms in the literature are outlined. Also, three cutting plane algorithms for solving (SRP) problems are proposed, which seem to be new and cannot be reduced to each other

Global Optimization for the Sum of Concave-Convex Ratios Problem

This paper presents a branch and bound algorithm for globally solving the sum of concave-convex ratios problem (P) over a compact convex set. Firstly, the problem (P) is converted to an equivalent problem (P1). Then, the initial nonconvex programming problem is reduced to a sequence of convex programming problems by utilizing linearization technique. The proposed algorithm is convergent to a global optimal solution by means of the subsequent solutions of a series of convex programming problems. Some examples are given to illustrate the feasibility of the proposed algorithm