Copositive optimization – Recent developments and applications

• Published in: European journal of operational research Vol. 216; no. 3; pp. 509 - 520
• Main Author: Bomze, Immanuel M.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.02.2012; Elsevier; Elsevier Sequoia S.A
• Subjects: Algorithms; Applied sciences; Clique number; Combinatorial analysis; Communities; Completely positive matrix; Computer science; control theory; systems; Conics; Convexity gap; Crossing number; Exact sciences and technology; Flows in networks. Combinatorial problems; Information retrieval. Graph; Mathematical models; Mathematical problems; Operational research; Operational research and scientific management; Operational research. Management science; Operations research; Optimization; Optimization algorithms; Robust optimization; Standard quadratic optimization; Studies; Theoretical computing; Versatility; Clique number; Robust optimization; Standard quadratic optimization; Convexity gap; Completely positive matrix; Crossing number; Stochastic model; Cone; conic programming; Graph clique; Local optimum; Minimization; Global optimum; Combinatorial optimization; Global local method; Deterministic model; NP hard problem; Optimality condition; Optimality criterion; Convexity; Mathematical programming
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2011.04.026

► Copositive optimization is an important problem class within conic optimization which receives much attention nowadays. ► It allows reformulation of diverse NP-hard problems, including nonconvex, mixed-binary, and fractional quadratic problems. ► Combinatorial applications: quadratic assignment, bounding the clique and the crossing number, graph coloring and partitioning. ► Applications in the continuous domain: stability in switched systems, optimality conditions, and tight convex underestimation. ► Further applications: rigid body mechanics, networks in queueing, traffic, reliability, uncertainty in Q2 mixed-integer LPs. Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.