Download e-book for iPad: A Reformulation-Linearization Technique for Solving Discrete by Hanif D. Sherali
By Hanif D. Sherali
This ebook bargains with the idea and purposes of the Reformulation- Linearization/Convexification strategy (RL T) for fixing nonconvex optimization difficulties. A unified remedy of discrete and non-stop nonconvex programming difficulties is gifted utilizing this procedure. 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 . should be equivalently J expressed because the polynomial constraint x . (1-x . ) = zero. the inducement for this publication is J J the position of tight linear/convex programming representations or relaxations in fixing such discrete and non-stop nonconvex programming difficulties. The imperative thrust is to start with a version that gives an invaluable illustration and constitution, after which to extra develop this illustration via computerized reformulation and constraint iteration ideas. As pointed out above, the point of interest of this booklet is the improvement and alertness of RL T to be used as an automated reformulation method, and likewise, to generate robust legitimate inequalities. The RLT operates in levels. within the Reformulation part, specific sorts of extra implied polynomial constraints, that come with the aforementioned constraints relating to binary variables, are appended to the matter. The ensuing challenge is consequently linearized, other than that convinced convex constraints are often retained in XV specific distinctive situations, within the Linearization/Convexijication part. this can be performed through the definition of appropriate new variables to exchange each one targeted variable-product time period. the better dimensional illustration yields a linear (or convex) programming relaxation.
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Additional info for A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems
Sherali and Myers (1989) also discuss and test various strategies and provide guidelines for composing suitable Lagrangian dual formulations. Second, an appropriate nondifferentiable optimization technique must be employed to solve the Lagrangian dual problem. Cutting plane or outer linearization methods (see Bazaraa and Goode, 1979) can quickly accumulate far too many constraints in the master program, and thereby get bogged down. The frrst-order subgradient methods of Poljak (1967, 1969), and Held et al.
5c). 3. For any d E (1, ... 5c). and for each (11 , J2 ) of order d). Let x be any binary vector. 23a) and A = nAx. w1 jeJ 1 for all k and vA Jk = 1, ... , 1 jeJ J ~:; N with I Jl = 1, ... , d. 23b) 44 Sherali and Adams Proof. ~(11 ,12 ) match those of Fd(J1 ,12 ) and ykF/11 ,12 ), respectively, and so (X, y, w, v) E zd with X binary. Conversely, let (x, y, w, v) E Zd. 23) holds true. Note that for d = 1, the set Z1 has constraints x. ) 1 ~ (yk - v 'k) 1 ~ 0 for j = 1, ... , n, k = 1, ... , m. Hence, 0::;; yk ::;; 1, and for any j = 1, ...
These factors are then used to multiply each of the constraints defining X (including the variable bounding restrictions), to create a (nonlinear) polynomial mixed-integer zero-one programming problem. relationship x 2 J = x J. , j J = 1, ... , and relaxing integrality, the nonlinear polynomial problem is re-linearized J into a higher dimensional polyhedral set Xd defmed in terms of the original variables Sherali and Adams 10 (x, y) and the new variables (w, v). For Xd to be equivalent to X, it is only necessary to enforce x to be binary valued, with the remaining variables treated as continuous valued, since the binariness on the x-variables is shown to automatically enforce the required product relationships on the w- and v-variables.
A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems by Hanif D. Sherali