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 ...

We present numerical results of a comparative study of codes for nonlinear and nonconvex mixed-integer optimization. The underlying algorithms are based on sequential quadratic programming (SQP) with stabilization by trust-regions, linear outer approximations, and branch-and-bound techniques. Themixed-integer quadratic programming subproblems are solved by a branch-and-cut algorithm. Second ord...

This paper introduces two fundamental families of ‘quasipolyhedra’ — polyhedra with a countably infinite number of facets — that arise in the context of integer quadratic programming. It is shown that any integer quadratic program can be reduced to the minimisation of a linear function over a quasi-polyhedron in the first family. Some fundamental properties of the quasi-polyhedra are derived, a...

Let (MQP ) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (MQP ), i.e. we reformulate (MQP ) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that ou...

The Perspective Relaxation (PR) is a general approach for constructing tight approximations to MixedInteger NonLinear Problems with semicontinuous variables. The PR of a MINLP can be formulated either as a Mixed-Integer Second-Order Cone Program (provided that the original objective function is SOCPrepresentable), or as a Semi-Infinite MINLP. In this paper, we show that under some further assum...

In computational complexity theory, a decision problem is NP-complete when it is both in NP and NP-hard. Although a solution to a NP-complete can be verified quickly, there is no known algorithm to solve it in polynomial time. There exists a method to reduce a SAT (Satifiability) problem to Subset Sum Problem (SSP) in the literature, however, it can only be applied to small or medium size probl...

this paper presents a framework for long term transmission expansion planning in competitive, electricity markets. transmission lines and phase shifters are taken into account as expansion options. maximization of the network users' benefits, with satisfying security constraints are considered as the criterion for transmission expansion planning. the elements of the objective function are the...

the markowitz’s optimization problem is considered as a standard quadratic programming problem that has exact mathematical solutions. considering real world limits and conditions, the portfolio optimization problem is a mixed quadratic and integer programming problem for which efficient algorithms do not exist. therefore, the use of meta-heuristic methods such as neural networks and evolutionar...