By Hanif D. Sherali

ISBN-10: 1441948082

ISBN-13: 9781441948083

ISBN-10: 1475743882

ISBN-13: 9781475743883

This publication offers with the speculation and functions of the Reformulation- Linearization/Convexification method (RL T) for fixing nonconvex optimization difficulties. A unified therapy of discrete and non-stop nonconvex programming difficulties is gifted utilizing this method. In essence, the bridge among those sorts of nonconvexities is made through a polynomial illustration of discrete constraints. for instance, the binariness on a 0-1 variable x . might be equivalently J expressed because the polynomial constraint x . (1-x . ) = zero. the incentive for this booklet is J J the function of tight linear/convex programming representations or relaxations in fixing such discrete and non-stop nonconvex programming difficulties. The significant thrust is to start with a version that offers an invaluable illustration and constitution, after which to additional improve this illustration via computerized reformulation and constraint iteration concepts. As pointed out above, the point of interest of this booklet is the advance and alertness of RL T to be used as an automated reformulation strategy, and in addition, to generate powerful legitimate inequalities. The RLT operates in levels. within the Reformulation part, specific sorts of extra implied polynomial constraints, that come with the aforementioned constraints when it comes to binary variables, are appended to the matter. The ensuing challenge is for this reason linearized, other than that convinced convex constraints are often retained in XV specific unique situations, within the Linearization/Convexijication section. this is often performed through the definition of appropriate new variables to exchange each one distinctive variable-product time period. the better dimensional illustration yields a linear (or convex) programming relaxation.

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**Extra info for A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems**

**Example text**

11 ), and the proof is complete. D The equivalence of X to Xd for any d E { 0, ... , n} under integrality restrictions on the x-variables, and the hierarchy among the relaxations are established next. 1. 6), ford = 0, 1, ... , n. 12) Inparticular, XPd n{(x,y): xbinary} Proof. Consider any dE =Xforall d = 0,1, ... ,n. {1, ... 5a) are also satisfied. Toward this end, consider any (11 , 1 2 ) of order (d -1), and any r E {l, ... ,R}. 5a) for Next, let us show that conv(X) any ( x, y) E conv(X) ~ ~ Xn.

D. 1) as equalities by using slack variables. In this chapter, we present the basic construction process of the ReformulationLinearization Technique (RLT). 1), given any level {0, ... , n}, this technique first converts the constraint set into a polynomial mixed- integer zero-one set of restrictions by multiplying the constraints with some suitable ~ degree polynomial factors involving the n binary variables and their complements, and subsequently linearizes the resulting problem through appropriate variable transformations.

To view their relaxations as Balas' hull relaxations requires manipulating the representation at any stage d to write it as a conjunction of a non-standard set of disjunctions. Moreover, by the nature of the RLT approach, Sherali and Adams also show (see Chapter 2) how one can readily construct a hierarchy of linear relaxations leading to the convex hull representation for mixed-integer zero-one polynomial programming problems having no cross-product terms among continuous variables, an important result not shared by the earlier disjunctive work.

### A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems by Hanif D. Sherali

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