Accurate modelling of real-world problems often requires nonconvex terms to be introduced in the model, either in the objective function or in the constraints. Nonconvex programming is one of the hardest fields of optimization, presenting many challenges in both practical and theoretical aspects. The presence of multiple local minima calls for the application of global optimization techniques. ...

An approach to nonlinear constrained mathematical programming problems which makes use of a second order derivative is presented. By using a second order modified objective function method, a modified optimization problem associated with a primal mathematical programming problem is constructed. This auxiliary optimization problem involves a second order approximation of an objective function co...

In the field of nonlinear programming (in continuous variables) convex analysis [22, 23] plays a pivotal role both in theory and in practice. An analogous theory for discrete optimization (nonlinear integer programming), called " discrete convex analysis " [18, 17], is developed for L-convex and M-convex functions by adapting the ideas in convex analysis and generalizing the results in matroid ...

Branch-and-Bound (B&B) is perhaps the most fundamental algorithm for the global solution of convex Mixed-Integer Nonlinear Programming (MINLP) problems. It is well-known that carrying out branching in a non-simplistic manner can greatly enhance the practicality of B&B in the context of Mixed-Integer Linear Programming (MILP). No detailed study of branching has heretofore been carried out for MI...

We study mixed integer nonlinear programs (MINLP)s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume fixed values, and, when it is “turned on”, forces them to belong to a convex set. Many practical MINLPs contain integer ...

Minimizing a convex function over a convex region is probably the core problem in the Nonlinear Programming literature. Under the assumption of the function of a differentiable function of interest, several methods have been proposed and successively studied. For example, the Steepest Descent Method consists of performing a line search along a descent direction given by minus the gradient of th...

Abstract This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an in...

abstract it is the purpose of this article to introduce a linear approximation technique for solving a fractional chance constrained programming (cc) problem. for this purpose, a fuzzy goal programming model of the equivalent deterministic form of the fractional chance constrained programming is provided and then the process of defuzzification and linearization of the problem is started. a samp...

In this paper we develop a subgradient optimization methodology for convex multiparametric nonlinear programs. We define a parametric subgradient, extend some classical optimization results to the multiparametric case, and design a subgradient algorithm that is shown to converge under traditional conditions. We use this algorithm to solve two illustrative example problems and demonstrate its ac...

This note presents the characterization of the stability set of the first kind for multiobjective nonlinear programming (MONLP) problems with fuzzy parameters either in the constraints or in the objective functions without any differentiability assumptions. These fuzzy parameters are characterized by triangular fuzzy numbers (TFNs). The existing results concerning the parametric space in convex...