By Hanif D. Sherali

ISBN-10: 1441948082

ISBN-13: 9781441948083

ISBN-10: 1475743882

ISBN-13: 9781475743883

This booklet offers with the idea and purposes 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 process. In essence, the bridge among those different types of nonconvexities is made through a polynomial illustration of discrete constraints. for instance, the binariness on a 0-1 variable x . could be equivalently J expressed because the polynomial constraint x . (1-x . ) = zero. the incentive for this booklet is J J the position 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 presents an invaluable illustration and constitution, after which to extra improve this illustration via computerized reformulation and constraint new release innovations. As pointed out above, the focus of this ebook is the improvement and alertness of RL T to be used as an automated reformulation approach, and likewise, to generate powerful legitimate inequalities. The RLT operates in levels. within the Reformulation section, particular types of extra implied polynomial constraints, that come with the aforementioned constraints in relation to binary variables, are appended to the matter. The ensuing challenge is therefore linearized, other than that sure convex constraints are often retained in XV specific distinctive circumstances, within the Linearization/Convexijication section. this can be performed through the definition of compatible new variables to switch each one specific variable-product time period. the better dimensional illustration yields a linear (or convex) programming relaxation.

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**Additional resources for A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems**

**Example text**

Chapter 7 discusses this procedure along with an extension due to Sherali (1996) to handle rational exponents on the polynomial functions, instances of which arise in many engineering design problems (see A Reformulation-Linearization Technique 17 Floudas and Pardalos, 1987, and Grossmann, 1996). Again, various transformations and (partial) constraint and bound factor products can be gainfully employed to judiciously construct an appropriate linear programming relaxation. For example, Shor (1990) and Floudas and Visweswaran (1991) have suggested a successive quadratic variable substitution strategy to transform a given polynomial programming problem to one of equivalently minimizing a quadratic objective function subject to quadratic constraints.

N}, using the following two steps that comprise the Reformulation-Linearization Technique (RLT). 2). Upon using the identity x 2 1 =x. (and so J x. , j = 1, ... , n, this gives the following set of additional, implied, J nonlinear constraints L [ jeJ 1 ar. : 0 for r = 1, ... : 0 for each (11, 12 ) of order D =min{d + 1, n} fork= l, ... ,m,andforeach (11 ,12 ) oforderd. 3) in expanded form as a sum of monomials, linearize them by substituting the following variables for the corresponding nonlinear terms for each 1 r;;;; N: w 1 = flx.

N, are taken J J J J time and are restricted to be nonnegative, where a is a at a the highest degree of any polynomial term appearing in the problem. The resulting problem is then linearized by substituting a variable X1 for each product term tr x . , where J might contain repeated jeJ J indices. Using this linear programming relaxation, and partitioning the problem based on splitting the bounding interval for that variable x p which produces the highest discrepancy between X Jv P and X1 over all J in the linear programming solution, a convergent branch-and-bound algorithm is designed.

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

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