M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems," Technical Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan, July 1998, to appear in Mathematical Programming.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....been shown that KSDP (F ) and K CQI (F ) are equivalent. Theorem 2. 1 ( 7, 22] KSDP (F ) K CQI (F ) As an immediate consequence of the theorem, we know that, if Q i O for i = 1; m, F = KSDP (F ) K CQI (F ) Another consequences are shown in [7, 22] In particular, Kojima and Tuncel [22, 23] used Theorem 2.1 to derive some features of their successive convex relaxation method for a very large class of optimization problems. 4 2.2 The Maximum Weight Stable Set Problem Let G = V; E) be an undirected graph. Then the maximum weight stable set problem (SSP) is formulated as follows : ....

M. Kojima and L. Tuncel, Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems, Research Report #341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Japan, July 1998.

....the Lagrangian relaxation approach for general quadratically constrained quadratic problems (Q 2 P ) In this Section we brie y outline the approach for the general Q 2 P and speci c instances are considered in some detail in Section 4. This general quadratic problem is also studied in e.g. [30, 67, 66, 68, 115] and [104, 73, 71, 83, 15] The more general polynomial optimization problem is considered in [74] which presents a relaxation very similar to SDP3 but motivated by result results in the theory of moments and positive polynomials. The quadratic problem we consider is the following Q 2 P : Q 2 ....

....to our earlier claim that adding redundant quadratic constraints strengthens the SDP relaxation. An excellent illustration of the e ectiveness of this strategy is presented in Section 4.4 where this approach achieves strong duality. Another approach is presented in detail in Kojima and Tun cel [67, 66]. For problems that also have linear equality constraints the notion of copositivity can be used to strengthen the SDP relaxation [102] However the result is not a tractable relaxation in general. 3.1 Solving SDPs arising from Q 2 Ps There are many existing packages for solving SDPs in the ....

M. KOJIMA and L. TUNCEL. Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems. Technical Report CORR98-34, Dept. of Combinatorics and Optimization, University of Waterloo, 1998.

....q k (y) means that we have a stronger dual. This can be phrased as adding redundant constraints to get new valid inequalities to strengthen the relaxation. We will see how this occurs when we look at orthogonally constrained problems below. Another approach is also specified in detail in [28] and [44, 43]. For problems that also have linear equality constraints, one can use the notion of copositivity to strengthen the SDP relaxation. However, this does not result in a tractable relaxation in general, see [82] Specific instances of these relaxations (quadratic assignment and graph partitioning ....

M. KOJIMA and L. TUNCEL. Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems. Technical Report CORR98-34, Dept. of Combinatorics and Optimization, University of Waterloo, 1998.

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M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems," Technical Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan, July 1998, to appear in Mathematical Programming.

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M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems", Mathematical Programming, 89 (2000) 79--111.

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M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems," Math. Programming 89 (2000) 79--111.

....Programming) relaxation method by Lov asz Schrijver [12] were regarded as their pioneering works. They had been modified, generalized and extended to various problems and methods; the RLT [17, 18] for 0 1 mixed integer polynomial programs, the SCRM (Successive Convex Relaxation Method) [7, 8] for QOPs (Quadratic Optimization Problems) the SDP relaxation [9, 10] of polynomial programs, and SOCP (Second Order Cone Programming) relaxations [5, 6] for QOPs. These methods share the following basic idea. i) Add (redundant) valid inequality constraints to a target optimization problem ....

....the relaxed lifted problem in the matrix space back to the original Euclidean space R . In some special cases such as the max cut problem [3] or some classes of 0 1 QOPs [6, 14, 22, 23, 25] i) is not included. but it is inevitable in general. In fact, i) is a key issue in the papers [1, 5, 7, 8, 9, 10, 12, 16, 17, 18] referred above; they employ various techniques of constructing effective valid inequality constraints to strengthen their convex relaxations. Theoretically, Lasserre s SDP relaxation method [9] is very powerful in the sense that optimal values of fairly general polynomial programs having a ....

[Article contains additional citation context not shown here]

M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems," Math. Programming 89 (2000) 79--111.

....function value of QOP (1) While the SCRMs proposed by [9] enjoy the global convergence property that # # k=0 C k = the convex hull of F , they involve an infinite number of semi infinite LPs or SDPs to generate a new convex relaxation C k of F . To resolve this di#culty, Kojima and Tuncel [10] proposed implementable versions of SCRMs by bringing two new techniques, discretization and localization , into their theoretical framework. Their techniques allow us to solve finitely many LPs or SDPs having finitely many inequality constraints, so that the discretized localized versions are ....

....some implementation details of discretized localized SCRMs and gave preliminary numerical results. We introduce a standard serial algorithm of discretized localized SCRMs in Section 2.1, and we present some basic properties on the algorithm in Section 2.2. 2. 1 Preliminaries The previous works [8, 9, 10, 24, 25] of SCRMs handled general quadratic optimization problems (abbreviated by QOPs) with the following form: max c T x : x # F , 2) where F # x # C 0 : qf(x; #, q, Q) # 0 (#qf( #, q, Q) # P F ) C 0 # a given compact convex set including F ; we assume that C 0 is represented in terms ....

[Article contains additional citation context not shown here]

M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems," Technical Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan, July 1998, to appear in Mathematical Programming.

....function value of QOP (1) While the SCRMs proposed by [9] enjoy the global convergence property that 1 k=0 C k = the convex hull of F , they involve an infinite number of semi infinite LPs or SDPs to generate a new convex relaxation C k of F . To resolve this difficulty, Kojima and Tuncel [10] proposed implementable versions of SCRMs by bringing two new techniques, discretization and localization , into their theoretical framework. Their techniques allow us to solve finitely many LPs or SDPs having finitely many inequality constraints, so that the discretized localized versions are ....

....some implementation details of discretized localized SCRMs and gave preliminary numerical results. We introduce a standard serial algorithm of discretized localized SCRMs in Section 2.1, and we present some basic properties on the algorithm in Section 2.2. 2. 1 Preliminaries The previous works [8, 9, 10, 24, 25] of SCRMs handled general quadratic optimization problems (abbreviated by QOPs) with the following form: maxfc T x : x 2 Fg; 2) where F j fx 2 C 0 : qf(x; fl; q; Q) 0 (8qf( Delta; fl; q; Q) 2 P F )g; C 0 j a given compact convex set including F ; we assume that C 0 is represented in terms of ....

[Article contains additional citation context not shown here]

M. Kojima and L. Tuncel, "Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems," Technical Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan, July 1998, to appear in Mathematical Programming.

.... programs such as 0 1 linear and quadratic integer programs [20] linear complementarity problems [4] bilinear matrix inequalities [11, 16] bilevel linear and quadratic programs [30] and sum of linear fractional programs [6, 24] A general QP can be represented by a problem in the following form [13, 14]. max c T x s.t. x 2 F: 1) Here c 2 R n j the n dimensional Euclidean space; a T : the transposition of a vector a 2 R n ; F j fx 2 R n : p(x) 0 (8p(1) 2 P F )g; P F Q j the set of quadratic functions on R n : Recently, Kojima and Tun cel [13] established a theoretical ....

....of semi infinite SDPs (or semi infinite LPs) to generate a new approximation C k of c.hull(F ) Hence, the implementation can not be practically conducted on a computer which is available nowadays. To resolve the difficulty, they proposed corresponding variants of the methods in a later paper [14] by bringing two new techniques, discretization and localization into the theoretical framework. The discretization makes it possible to approximate an infinite number of semi infinite SDPs (or semi infinite LPs) which need to be solved at each iteration of the algorithm by a finite number ....

[Article contains additional citation context not shown here]

M. Kojima and L. Tun¸cel, (1998), "Discretization and Localization in Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems," Technical Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo, Japan.

....3. All the discussion will consider the BMIEP, though it will also be valid for any subproblem of the branch and bound algorithms. 2. 1 Convex Relaxation of the BMIEP Convex (linear) relaxation is one of the techniques frequently utilized in 0 1 integer programming and nonlinear programming [18, 19, 23, 25, 34]. Several authors have utilized this technique to obtain an LMI (SDP) relaxation of the BMIEP [8, 14, 31] Their approach requires the following equivalent formulation for the BMIEP. Consider the BMIEP (2) and the matrix function (1) Letting w ij = x i y j (i 2 I n ; j 2 J m ) we obtain: ....

....the proposed algorithms to solve these problems. A different approach to solve problems involving BMIs was recently proposed by Kojima and Tun cel. They announced two conceptual algorithms based on LP and SDP relaxations to solve nonconvex quadratic programs [18] and their implementable variants [19]. In their paper, they showed that their framework is quite general to be applied to Quadratic Matrix Inequalities (QMIs) which includes BMIs. Jarre [16] proposed an algorithm based on interior point method, SQP method and trust regions to solve SDP with nonlinear equalities. Some preliminary ....

M. Kojima and L. Tun¸cel, "Discretization and localization in successive convex relaxation method for nonconvex quadratic optimization problems," Research Report B-341, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, July 1998.

No context found.

M. Kojima and L. Tuncel. Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems. Mathematical Programming, Series A, 89:79--111, 2000.

No context found.

M. KOJIMA and L. TUNCEL. Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems. Technical Report CORR98-34, Dept. of Combinatorics and Optimization, University of Waterloo, 1998.

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