Uncorrected Proof 2008-01-18 ✐ Kao: Encyclopedia of Optimization — Entry 115 — 2008/1/18 — 19:46 — page 1 — LE-T E X ✐ ✐ ✐ Multi-Quadratic Integer Programming: Models and Applications 1 Multi-Quadratic Integer Programming: Models and Applications W. ART CHAOVALITWONGSE 1 ,XIAOZHENG HE 2 , 1 ANTHONY CHEN 3 2 1 Department of Industrial and Systems Engineering, 3 Rutgers University, Piscataway, USA 4 2 Department of Civil Engineering, University 5 of Minnesota, CE2 , USA 6 3 Department of Civil and Environmental Engineering, 7 Utah State University, CE2 , USA 8 MSC2000: CE3 9 Article Outline 10 Keywords and Phrases 11 Introduction 12 Multi-Quadratic Integer Program 13 Applications 14 Bilinear Problem 15 Minimax Problem 16 Mixed Integer Problem 17 Solution Techniques 18 Linear Forms of MQIP 19 References 20 Keywords and Phrases 21 Quadratic programming; Quadratic constraints; 22 Mixed-integer program; Linearization 23 Introduction 24 In this contribution, we consider Multi-Quadratic Pro- 25 gramming (MQP) problems, where the objective func- 26 tion is a quadratic function and the feasible region is 27 defined by a finite set of quadratic and linear con- 28 straints. They can be formulated as follows: 29 min x T Qx + c T x s.t. x T A j x + B j x  b j ; j =1;:::;m x  0 ; (1) 30 where A j is an (n  n) matrix corresponding to the mth 31 quadratic constraint, and B j is the jth row of the (m  n) 32 matrix B. 33 MQP plays an important role in modeling many diverse 34 problems. The MQP encompasses many other opti- 35 mization problems since it provides a much improved 36 model compared to the simpler linear relaxation of 37 a problem. Indeed, linear mixed 0–1, fractional, bilin- 38 ear, bilevel, generalized linear complementarity, and 39 many more programming problems are or can easily be 40 reformulated as special cases of MQP. However, there 41 are theoretical and practical difficulties in the process of 42 solving such problems. However, very large linear mod- 43 els can be solved efficiently; whereas MQP problems 44 are in general NP -hard and numerically intractable. 45 The problem of finding a feasible solution is also 46 NP -hard. This is because MQP is a generalization of 47 the linear complementarity problem [28]. The nonlin- 48 ear constraints in MQP define a feasible region which 49 is in general neither convex nor connected. Moreover, 50 even if the feasible region is a polyhedron, optimizing 51 the quadratic objective function is strongly NP -hard 52 as the resulting problem is considered to be the dis- 53 joint bilinear programming problem. Therefore, finding 54 a finite and exact algorithm that solves large MQP prob- 55 lems is impractical. Even for the convex case (when 56 Q and A j are positive semidefinite), there are very few 57 algorithms for solving MQP problems. However, the 58 MQP constitutes an important part of mathematical pro- 59 gramming problems, arising in various practical appli- 60 cations including facility location, production planning, 61 VLSI chip design, optimal design of water distribution 62 networks, and most problems in chemical engineering 63 design. 64 The MQP was first introduced in the seminal paper of 65 Kuhn and Tucker [30]. Later on, the case of MQP with 66 a single quadratic constraint in the problem was dis- 67 cussed in [54,56]. The first general approach for solving 68 MQP problems was proposed in [11], where the follow- 69 ing two Lagrange functions for MQP are considered: 70 L 1 (x;)= x T Qx + c T x + m X j =1  j (x T A j x  B j x  b j ) ; L 2 (x;;)= L 1 (x;)   i x i ; 71 where  and  are the multipliers for the quadrat- 72 ic and bound constraints respectively. A cutting plane 73 algorithm was applied to solve this problem; that is, 74 the algorithm solves a sequence of linear master prob- 75 lems that minimize a piecewise linear function con- 76 structed from the Lagrange functions for constant x, and 77 a primal problem with either an unconstrained quadrat- 78 ic function (using L 2 (x;;)) or a quadratic function 79 over the nonnegative orthant (using L 1 (x;)) [20]. 80 Please note that pagination and layout are not final. CE2 Please provide city. CE3 Please provide MSC number. Editor’s or typesetter’s annotations (will be removed before the final T E X run)