(253a) Enhancing Relaxations for Nonconvex Mixed-Integer Quadratically-Constrained Quadratic Programs
AIChE Annual Meeting
2018
2018 AIChE Annual Meeting
Computing and Systems Technology Division
Advances in Deterministic Global Optimization
Tuesday, October 30, 2018 - 8:00am to 8:19am
Mixed-integer quadratically-constrained quadratic programs (MIQCQPs), and related classes of problems such as quadratically-constrained quadratic programs (QCQPs) and quadratic programs (QPs) arise in a wide variety of engineering and scientific applications including pooling and blending in process networks [12], facility location [23], production planning [8], quadratic assignment [9] and portfolio optimization [15]. Due to their practical importance, these classes of problems have been extensively studied in the mathematical programming literature. They are typically solved using spatial branch-and-bound algorithms, whose efficiency primarily depends on the quality of the relaxations constructed to bound the problem.
Commonly used relaxations for these type of problems can be classified into four groups. The first group involves relaxations obtained via standard factorable programming techniques whereby each quadratic function is decomposed iteratively by introducing intermediate variables and constraints for each nonconvex term, until each intermediate expression can be outer-approximated by its convex and/or concave envelopes [10, 20, 22]. While these techniques are relatively simple to implement, they typically result in large relaxation gaps. The second group is given by semi-definite programming (SDP) relaxations. This approach has received significant attention in recent years [1, 5, 6, 21]. While SDP relaxations often provide strong bounds, they are computationally expensive to solve, which limits their use within general purpose branch-and-bound global solvers. The third group relies on the reformulation-linearization techniques (RLT) [17â19]. Though these techniques typically yield tight bounds, the resulting relaxations grow quickly in size, and thus can become expensive to solve. The fourth group consists of factorable relaxations augmented by adding various classes of valid inequalities such as SDP-based cuts [7, 16], RTL-based cuts [24, 25], facets of the envelopes of edge-concave and multilinear subexpressions [2, 3, 11, 13], and mixed-integer cuts [4]. These augmented relaxations provide stronger bounds than standard factorable relaxations while remaining computationally inexpensive to solve. Due to its practicality, this class of relaxations is employed in most general-purpose global optimization packages.
In this work, we show how certain facets of the Boolean Quadric Polytope [14] can be utilized in order to enhance factorable relaxations of MIQCQPs constructed by global optimization solvers. To illustrate the benefits of this approach, we propose a computational implementation which integrates these facets as cutting planes into the branch-and-reduce global solver BARON. Our implementation consists of several algorithmic developments including tools for the identification of substructures in a graph representation of the problem, as well as efficient cut generation and separation algorithms which are embedded at every node of the branch-and-bound tree. We demonstrate the impact of the proposed implementation by conducting an extensive computational study on a large collection of problems selected from publicly available test sets.
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