One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over … Read more
This work extends the ones of Hu et al. (2008) and Bai et al. (2013) of a logical Benders approach for globally solving Linear Programs with Complementarity Constraints. By interpreting the logical Benders method as a reversed branch-and-bound method, where the whole exploration procedure starts from the leaf nodes in an enumeration tree, we provide … Read more
In this paper, we study some bounds for nonconvex quadratically constrained quadratic programs. Recently, Zamani has proposed a dual for linearly constrained quadratic programs, where Lagrange multipliers are affine functions. By using this method, we propose two types of bounds for quadratically constrained quadratic pro- grams, quadratic and cubic bounds. For quadratic bounds, we use … Read more
Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The … Read more
We study sets defined as the intersection of a rank-1 constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-1 set is polyhedral or second-order cone representable. In all these cases, we also show that a linear objective can be … Read more
Convexification is a core technique in global polynomial optimization. Currently, two different approaches compete in practice and in the literature. First, general approaches rooted in nonlinear programming. They are comparitively cheap from a computational point of view, but typically do not provide good (tight) relaxations with respect to bounds for the original problem. Second, approaches … Read more
The purpose of this paper is to find appropriate solutions to concave quadratic programming using outer approximation algorithm, which is one of the algorithm of global optimization, in the target of the strong of concavity of object function i.e. high of nonlinear risk of portfolio. Firstly, my target model is a mathematical optimization programming to … Read more
We present a framework to obtain valid inequalities for optimization problems constrained by a reverse convex set, which is defined as the set of points in a polyhedron that lie outside a given open convex set. We are particularly interested in cases where the closure of the convex set is either non-polyhedral, or is defined … Read more
A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global optimality is a well-known NP-hard problem and a traditional approach is to use convex relaxations and branch-and-bound algorithms. This paper makes a … Read more
Given a factorable function f, we propose a procedure that constructs a concave underestimor of f that is tight at a given point. These underestimators can be used to generate intersection cuts. A peculiarity of these underestimators is that they do not rely on a bounded domain. We propose a strengthening procedure for the intersection … Read more