A strong conic quadratic reformulation for machine-job assignment with controllable processing times

Abstract

a b s t r a c tWe describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation.

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.…”

“…It is therefore not surprising that (PRel) can be written as a SOCP, as recently proposed in [1,9] following suggestions dating back to [15], provided that the same is possible for (MINLP). The reformulation of (PRel) as a SOCP is actually quite simple in the quadratic case…”

“…This can be more efficient than attacking (MINLP) directly [1,9]. We call the above the Conic Program (CP) reformulation.…”

“…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.…”

“…However, an issue with (PRel) is the high nonlinearity in the objective function due to the added fractional term. Two workable reformulations of (PRel) have been proposed: one as a Second-Order Cone Program (SOCP) [15,1,9], and the other as a Semi-Infinite Linear Program [5]. These are recalled in Section 2.…”