scholarly journals Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

2010 ◽  
Vol 130 (2) ◽  
pp. 359-413 ◽  
Author(s):  
Anureet Saxena ◽  
Pierre Bonami ◽  
Jon Lee
Keyword(s):  
Mixed Integer ◽  
Convex Relaxations ◽  
Quadratically Constrained

scholarly journals Convex relaxations for mixed integer predictive control

Automatica ◽  
2010 ◽  
Vol 46 (9) ◽  
pp. 1540-1545 ◽  
Author(s):  
Daniel Axehill ◽  
Lieven Vandenberghe ◽  
Anders Hansson
Keyword(s):  
Predictive Control ◽  
Mixed Integer ◽  
Convex Relaxations

Mixed-integer nonlinear optimization

Acta Numerica ◽  
2013 ◽  
Vol 22 ◽  
pp. 1-131 ◽  
Author(s):  
Pietro Belotti ◽  
Christian Kirches ◽  
Sven Leyffer ◽  
Jeff Linderoth ◽  
James Luedtke ◽  
...  
Keyword(s):  
Branch And Bound ◽  
Convex Functions ◽  
State Of The Art ◽  
Small Sample ◽  
The State ◽  
Decision Problems ◽  
Mixed Integer ◽  
Feasible Region ◽  
Convex Relaxations ◽  
Single Tree

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

scholarly journals Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

2015 ◽  
Vol 158 (1-2) ◽  
pp. 235-266 ◽  
Author(s):  
Alain Billionnet ◽  
Sourour Elloumi ◽  
Amélie Lambert
Keyword(s):  
Mixed Integer ◽  
Constrained Problems ◽  
Quadratically Constrained
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