Data networks
A recursive procedure to generate all cuts for 0-1 mixed integer programs
Mathematical Programming: Series A and B
A lift-and-project cutting plane algorithm for mixed 0-1 programs
Mathematical Programming: Series A and B
An improved branch and bound algorithm for mixed integer nonlinear programs
Computers and Operations Research
Computational study of a family of mixed-integer quadratic programming problems
Mathematical Programming: Series A and B
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
Cuts for mixed 0-1 conic programming
Mathematical Programming: Series A and B
Mathematical Programming: Series A and B
Perspective cuts for a class of convex 0–1 mixed integer programs
Mathematical Programming: Series A and B
Mixed-integer nonlinear programming: some modeling and solution issues
IBM Journal of Research and Development - Business optimization
An algorithmic framework for convex mixed integer nonlinear programs
Discrete Optimization
Projected Perspective Reformulations with Applications in Design Problems
Operations Research
Eigenvalue techniques for convex objective, nonconvex optimization problems
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
The chvátal-gomory closure of an ellipsoid is a polyhedron
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Mixed-integer nonlinear programs featuring "on/off" constraints
Computational Optimization and Applications
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We study mixed integer nonlinear programs (MINLP) 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 a fixed value, and, when it is "turned on", forces them to belong to a convex set. Most of the integer variables in known MINLP problems are of this type. We first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints. This is an interesting set that appears as a substructure in many applications. We present the convex hull description of this set. We then extend this to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions. Further, we show that for many classes of problems, the convex hull can be expressed via conic quadratic constraints, and thus relaxations can be solved via second-order cone programming. Our work is closely related with the earlier work of Ceria and Soares (1996) as well as recent work by Frangioni and Gentile (2006) and, Aktürk, Atamtürk and Gürel (2007). Finally, we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role. In particular, we present strong computational results with two applications - quadratic facility location and network design with congestion - that show the power of the reformulation technique.