Download e-book for kindle: Convexification and Global Optimization in Continuous and by Mohit Tawarmalani

By Mohit Tawarmalani

ISBN-10: 1441952357

ISBN-13: 9781441952356

ISBN-10: 1475735324

ISBN-13: 9781475735321

Interest in restricted optimization originated with the straightforward linear seasoned­ gramming version because it was once useful and maybe the single computationally tractable version on the time. limited linear optimization types have been quickly followed in different software parts and are probably the main prevalent mathematical types in operations learn and administration technological know-how on the time of this writing. Modelers have, even though, stumbled on the belief of linearity to be overly restrictive in expressing the real-world phenomena and difficulties in economics, finance, company, conversation, engineering layout, computational biology, and different parts that often call for using nonlinear expressions and discrete variables in optimization types. either one of those extensions of the linear programming version are NP-hard, therefore representing very tough difficulties. at the brighter facet, contemporary advances in algorithmic and computing expertise give the chance to re­ stopover at those issues of the desire of fixing essentially proper difficulties in moderate quantities of computational time. preliminary makes an attempt at fixing nonlinear courses focused on the de­ velopment of neighborhood optimization tools making certain globality less than the idea of convexity. nevertheless, the integer programming liter­ ature has targeting the advance of equipment that be sure worldwide optima. the purpose of this booklet is to marry the developments in fixing nonlinear and integer programming types and to strengthen new leads to the extra basic framework of mixed-integer nonlinear courses (MINLPs) with the objective of devising virtually effective international optimization algorithms for MINLPs.

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Extra resources for Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications

Example text

Otherwise, the feasible region is subdivided into partition elements. These partition elements are appended to a list of partition elements (open nodes) which need to be explored further in order to locate an optimal solution and verify its globality. A selection rule is applied to choose one partition element from the list of open nodes and a new relaxation is constructed taking advantage of its now reduced size. d). c) . A bounding scheme is called consistent if any unfathomed partition element can be further refined and, for any sequence of infinitely decreasing partition elements, the upper and lower bounds converge in the limit.

Then, the tightest convex relaxation of K, restricted to X x C 2 is the tightest convex relaxation over C 1 x C2 . This 40 CHAPTER 2. 17 since we have demonstrated a set C x for each point (Xl, X2), Xl ~ X such that (Xl, X2) ~ G~t(f\;). 19 (Rikun 1997). A function L(Xb ... ,Xk) is said to be a general multilinear function if for each i = 1, ... ,k the function L (x~ , ... ,Xi, ... ,x2) linearly depends on vector Xi provided all the other k - 1 vector arguments are fixed. The polyhedrality of the convex envelope of a general multilinear function over Cartesian product of polytopes was shown by Rikun (1997).

E. FUNCTIONS 39 Proof. C<=) It follows easily by taking X = C. (==}) Let us denote the epigraph of x and that over C by c. Since X ~ C, x ~ c. Taking the convex hull on both sides, it follows that conv(x) ~ conv(c). 3 that Xo E Gt(c)). Let us assume that (xo, x)) and (xo, c)). Then, there exist two points (X1,

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Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications by Mohit Tawarmalani

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