Mixed-integer Quadratic Programming is in NP

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP and integer linear programming is in NP

An upper bound on the number of non-unique assignments in relaxed (mixed) integer linear programs of the assignment type

An upper bound is given for the number of non-unique assignments when solving the linear programming relaxation of (mixed) integer linear programming problems in which the integer variables are governed by assignment type constraints. Key-words: (mixed) integer linear programming, assignment problems, class-room scheduling

Contemporary Approaches to the Solution of the Integer Programming Problem

The purpose of this thesis is to provide analysis of the modem development of the methods for solution to the integer linear programming problem. The thesis simultaneously discusses some main approaches that lead to the development of the algorithms for the solution to the integer linear programming problem. Chapter 1 introduces the Generalized Linear Programming Problem alongside with the properties of a solution to the Linear Programming Problem. The simplex procedure presented to solve the Linear Programming Problem by adding slack variables along with the artificial-basis technique. Chapter 2 refers to the primal-dual simplex procedure. The dual simplex algorithm reflects the dual simplex procedure. Chapter 3 discusses the mixed and alternative formulations of the integer programming problem. Chapter 4 considers the optimality conditions with the imposed relaxations to solve the Linear Programming Relaxation Problem. The methods of the Integer Programming are introduced for the Linear Programming Relaxation. Chapter 5 discusses the concepts of the Branch-and Bound method followed by the direct application of the Branch-and-Bound method. Chapter 6 introduces the fundamental concepts of the cutting method. The main concept of the valid inequalities presented for the Linear Programming Problem as well as for the Integer Programming Problem. Gomory\u27s Fractional cutting plane Algorithm represents the desired step to obtain the solution for the Integer Programming Problem. Furthermore, the mixed integer cuts generalizes the concepts to provide the corresponding solution for the Integer Programming Problem. Chapter 7 describes the Gomory method for the pure Integer Program followed by the Gomory method for the mixed Integer Program. In the Appendix the computer program LINDO is used. Throughout the whole thesis this computer program is applied to emphasize the very helpful tool in Linear Programming. All above mentioned chapters include the variety of examples corresponding to the Linear Programming Problem and the Integer Program

Mixed Integer Linear Programming Formulation Techniques

A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques. However, in some cases the resulting MIP can be either too weak or too large to be effectively solved by state of the art solvers. In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems

Alternative mathematical programming formulations for FSS synthesis

A variety of mathematical programming models and two solution strategies are suggested for the problem of allocating orbital positions to (synthesizing) satellites in the Fixed Satellite Service. Mixed integer programming and almost linear programming formulations are presented in detail for each of two objectives: (1) positioning satellites as closely as possible to specified desired locations, and (2) minimizing the total length of the geostationary arc allocated to the satellites whose positions are to be determined. Computational results for mixed integer and almost linear programming models, with the objective of positioning satellites as closely as possible to their desired locations, are reported for three six-administration test problems and a thirteen-administration test problem