SDP diagonalizations and perspective cuts for a class of nonseparable MIQP

Abstract

We present a new approach, requiring the solution of a SemiDefinite Program, for decomposing the Hessian of a nonseparable Mixed-Integer Quadratic problem to permit using perspective cuts to improve its continuous relaxation bound. The new method favorably compares with a previously proposed one requiring a minimum eigenvalue computation.

A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes

“…This results in (5) and (1) coincident for u ∈ {0, 1} n , hence (PRef) is a "good" reformulation of (MINLP) since its continuous relaxation, called the Perspective Relaxation (PRel), provides significantly stronger bounds than the continuous relaxation of (MINLP) [5,6,1,8,9]. We remark that u i f i (p i /u i ) for u i ≥ 0 is the perspective function of f i (p i ), a well-known tool in convex analysis, hence the name.…”

“…This procedure can easily be implemented by using the standard tools made available by off-the-shelf solvers such as Cplex. Again, this is usually more efficient than approaching (MINLP) directly [5,6,8].…”

“…This can be dodged with a reformulation trick first proposed in [5], and somewhat reminiscent of the so-called Larangian Decomposition; compute a diagonal matrix D 0 such that R = Q − D 0, change the objective function to p T Dp + z T Rz, and add the additional constraint z = p. In this way, the perspective reformulation can be applied to the-now, separable-p variables, while all the "nonseparability" in the objective function is moved to the "other" variables z. An efficient and effective way for computing a "large" D is by solving a single SemiDefinite Program [6].…”

“…Semi-continuous variables are very often found in models of real-world problems such as production planning problems [17,5,7,8], financial trading and planning problems [10,6], and many others [4,1,9]. These are variables that are constrained to either assume the value 0, or to lie in some given convex compact set P ⊆ R m ; in our applications P will always be a polyhedron.…”

Transforming Mathematical Models Using Declarative Reformulation Rules

“…Some are limited to syntactic reformulations, i.e., those that can be obtained by application of algebraic rewriting rules to the elements of a given model [11]. These reformulations are capable of exploiting syntactical structure of the model, such as presence of particular algebraic terms in parts of its algebraic description [7]. While being very relevant, these do not include all transformations that have shown to be of practical use.…”