Copositive Programming – a Survey

Abstract

Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.

Copositive optimization – Recent developments and applications

“…While the time may not yet be ripe for writing up the final standard text book in this domain, several authors nonetheless bravely took the challenge of providing an overview, thereby aiming at a rapidly moving target. A recent survey on copositive optimization is offered by [57], while [77] and [74] provide reviews on copositivity with less emphasis on optimization. Bomze [16] and Busygin [37] provided entries in the most recent edition of the Encyclopedia of Optimization.…”

Copositivity and constrained fractional quadratic problems

“…Comparing the resulting pair in (24) to the original CP formulation in (14) and in (17), we obtain a dimension reduction from n+1 as in the general CFQP case to n in the StFQP case. Based on this formulation, we proceed to lower bounds based on the SDP relaxation of (24).…”

“…Based on this formulation, we proceed to lower bounds based on the SDP relaxation of (24). Let ψ = max {λ : C − λB ∈ C n } .…”

“…The success of this topic is due, not only to the elegance of the theory, but also to the good results obtained in tighter semidefinite relaxations for hard combinatorial optimization problems. For recent papers with a survey flavor see, e.g., [10,19,24], and for a clustered bibliography [15].…”

On the irreducibility, self-duality an non-homogeneity of completely positive cones

“…When C = R n , both cones K C and E C are equal to S n + (the cone of positive semidefinite matrices), which is the underlying cone in semidefinite programming [23] and semidefinite linear complementarity problems [17], [18]. In the case of C = R n + (the nonnegative orthant), these cones reduce, respectively, to the cones of completely positive matrices and copositive matrices which have appeared prominently in statistical and graph theoretic literature [4] and in copositive programming [11]. In a path-breaking work, Burer [5] showed that a nonconvex quadratic minimization problem over the nonnegative orthant with some additional linear and binary constraints can be reformulated as a linear program over the cone of completely positive matrices.…”