By Hanif D. Sherali

This ebook bargains with the idea and purposes of the Reformulation- Linearization/Convexification procedure (RL T) for fixing nonconvex optimization difficulties. A unified therapy of discrete and non-stop nonconvex programming difficulties is gifted utilizing this strategy. 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 . might be equivalently J expressed because the polynomial constraint x . (1-x . ) = zero. the incentive for this publication is J J the function of tight linear/convex programming representations or relaxations in fixing such discrete and non-stop nonconvex programming difficulties. The relevant thrust is to start with a version that provides an invaluable illustration and constitution, after which to extra improve this illustration via automated reformulation and constraint iteration suggestions. As pointed out above, the point of interest of this e-book is the advance and alertness of RL T to be used as an automated reformulation strategy, and likewise, to generate robust legitimate inequalities. The RLT operates in levels. within the Reformulation part, particular types of extra implied polynomial constraints, that come with the aforementioned constraints on the subject of binary variables, are appended to the matter. The ensuing challenge is hence linearized, other than that convinced convex constraints are often retained in XV specific unique circumstances, within the Linearization/Convexijication part. this can be performed through the definition of compatible new variables to exchange every one targeted 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**

Ym ). Given a value of d set E of bounded continuous variables {1, ... , n}, this RLT procedure constructs various polynomial factors of degree d comprised of the product of some d binary variables x j or their complements (1 - x j). 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).

M. = J L J~N:jeJ UJfor j = 1, ... ,n and yk = L J~N U~ fork= 1, ... ,m. Proof. 16) equals 0 whenever H :1:- 0, and equals 1 when H = 0. 13b) must be satisfied, yielding a unique solution. 13). 14b). 13b) for J = {j}, j = 1, ... , n, k = 1, ... , m, the proof is complete. 3. 14), for example. Then for any k e {1, ... 14) is as follows: - uk123' vl23k - _ uk VIk- v3k I+ -- uk3 + - uk12 vi2k - uk 12 + uk13 + uk 13 + uk23 + + uk123' uk vi3k = _ uk 123' V2k- uk123' uk13 + k uk123' k k 2 +UI2 +U23 +UI23' + uk123' 39 A Reformulation-Linearization Technique v0k - = yk = uk 0 + uk 1 + uk + 2 uk 3 uk + + 12 uk l3 + uk 23 + uk 123" The following theorem now provides the desired convex.

M, (11 ,12 ) k of order (d+1) imply that fd(J1,J2 ) ~ am f;(J 1,J2 ) ~ 0 for all = 1, ... , m, and (11, 12 ) of order d. Proof. Consider any (11 , 12 ) of order d with 0 S: d < nand any k e {1, ... , m}, and let t e N- (11 u J2 ). 11) f~+I(JI + t, 12) + f~+l(11, 12 + t) = f~(JI, 12). 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, ...