M.V. RAMANA. An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....community who have completed so much successful work on linear programming. At the moment, interior point methods are the most successful algorithms for general SDP problems, see e.g. the above survey articles as well as the books [30] 46] and the recent theses [4] 1] 22] 35] 17] 32] [33]. The above references provide some evidence of the current high level of research activity in these areas. The main contribution of this paper is a new approach to solving EDMCP. This approach is different than those in the literature in two ways. First we change the EDMCP into an approximation ....

M.V. RAMANA. An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

....thickness sheets [14] and minimal compliance design with optimized materials [13] For these problems the matrices A i are no longer dyadic. 3 A direct proof is included to make the presentation more self contained, but this result and related proofs on bordering of matrices may be found in [12]. 5 Assume the augmented matrix positive definite. First consider nonzero h 1 2 IR and h 2 2 IR n , so that (h 1 ; h T 2 ) 2 6 4 f T f A 3 7 5 0 B h 1 h 2 1 C A 0 or h T 2 Ah 2 2h T 2 fh 1 h 2 1 0: In this case (h 1 6= 0) we can assume h 1 = 1 w.l.o.g. to obtain h ....

M.V. Ramana. An algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, PhD Thesis, John Hopkins University, Baltimore, MD 21218, October (1993)

....the n dimensional Euclidean space R n . Third, and also most importantly, we can apply the semidefinite programming (SDP) relaxation, which was originally developed for 0 1 integer programming problems by Lovasz and Schrijver [12] and later extended to nonconvex quadratic optimization problems [6, 18, 19], to the entire class of maximization problems having a linear objective function and finitely or infinitely many quadratic inequality constraints. See also [1, 8, 9, 13, 15, 23, 24, 29] In addition to the reasons above, we should mention that the maximization problem with a linear objective ....

....that x # F (P) and F (P) # F (P) We also see that F (P) is convex. Hence c.hull(F (P) # F (P) The SDP relaxation was originally proposed for combinatorial optimization problems and 0 1 integer programming problems [12] and later extended to quadratic optimization problems. See [1, 6, 8, 9, 15, 19, 18, 23, 24, 29]. 3. Main results. Now we are ready to describe our method for approximating a quadratic inequality constrained compact feasible region F of the minimization problem (1.1) Before running the method, we need to fix a semi infinite quadratic SUCCESSIVE CONVEX RELAXATIONS OF NONCONVEX SETS 757 ....

M. V. Ramana, An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, Ph.D. Thesis, Johns Hopkins University, Baltimore, MD, 1993.

....hold, the optimal sets X opt and Z opt are nonempty. For a proof, see Nesterov and Nemirovsky [NN94, x4.2] or Rockafellar [Roc70, x30] Theorem 1 is an application of standard duality in convex analysis, so the constraint qualification is not surprising or unusual. Wolkowicz [Wol81] and Ramana [Ram93, Ram95, RG95] have formulated two different approaches to a duality theory for semidefinite programming that does not require strict feasibility. For our present purposes, the standard duality outlined above will be sufficient. Assume the optimal sets are nonempty, i.e. there exist feasible x and Z with c ....

M. Ramana. An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems. PhD thesis, The Johns Hopkins University, 1993.

....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 P x ) q : max q 0 ....

M.V. RAMANA. An Algorithmic Analysis of Multiquadratic and Semidenite Programming Problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

....discussed above are equal to the optimal value of the Lagrangian dual of the equivalent program PE : 2.2. General Q 2 P We now move on to applying the Lagrangian relaxation to general quadratic constrained quadratic problems, denoted Q 2 P . The general Q 2 P problem is also studied in e.g. [28, 44, 84, 50, 48, 59, 51]. Quadratic bounds using a Lagrangian relaxation have been extensively studied and applied in the literature, for example in [45] and, more recently, in [46] The latter calls the Lagrangian relaxation the best convex bound . Discussions on Lagrangian relaxation for nonconvex programs also ....

M.V. RAMANA. An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

....(3) It must be underlined that the idea of formulating non convex control problems as an LMI problem with a rank constraint is not new, see for instance [9] Beyond the scope of systems control, rank constrained LMI problems also occur in mathematical programming and combinatorial optimization. In [25, 23] it is shown that all of the problems with polynomial objective and polynomial constraints (including quadratic programming, binary programming, integer programming and job shop scheduling, to mention just a few) may actually be formulated as rank one LMI optimization problems. 3 Linearization ....

....devoted to the study of algorithms for solving the rank one LMI problem. It should be insightful to assess the applicability to our problem of the various rank minimization heuristics proposed in the literature. For instance, the cutting plane technique for rank one linear programming proposed in [25] may prove useful. Recent theoretical results on the rank minimization problem [22] also deserve to be considered. Finally, it should also be highly instructive to study deterministic global optimization techniques [12, 18, 5, 2] that can take advantage of the special structure of the rank one LMI ....

M. V. Ramana, "An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems", Ph. D. Thesis, The Johns Hopkins University, Baltimore, Maryland, 1993.

....problems that can be approximately solved in polynomial time. The complexity is based on self concordant barriers for the cone P: The smallest (best) barrier parameter is given in [25] The complexity of determining the (exact) feasibility of a system of linear matrix inequalities is discussed in [49, 52]. 3 ALGORITHMS 3.1 Interior Point Algorithms The SDP research is highlighted by the elegant primal dual interior point algorithms. The interior point methods are extensions of polynomial time algorithms developed originally for LP. The seminal work in [42] shows that log det(X) is a ....

M.V. RAMANA. An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

....as a QOP (1) after an appropriate transformation. 4.2 Implementations For the comparison, we implemented Algorithm DLSSILP1 with two different P F and P k , and two other typical algorithms related with the lifting and project procedure for QOP. The latter two algorithms are the SDP relaxation [1, 3, 8, 10, 17, 20, 21, 27, 31] and the LP relaxation proposed by Sherali and Tuncbilek [25, 26] We call them Algorithms DLSSILP1, DLSSILP1 Sherali(LP) Sherali(SDP) and Sherali(LP) respectively. In the transformed QOP form such as (7) 8) and (9) there are usually linear constraints in addition to quadratic ones. The input ....

M. V. Ramana (1993), An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, MD.

....as a QP (1) after an appropriate transformation. 4.2 Implementation For the comparison, we implemented Algorithm DLSSILP1 with two different P F and P k , and two other typical algorithms related with the lifting and project procedure for QP. The latter two algorithms are the SDP relaxation [1, 3, 8, 10, 17, 20, 21, 27, 31] and the LP relaxation proposed by Sherali and Tuncbilek [25, 26] We call them Algorithms DLSSILP1, DLSSILP1 Sherali(LP) Sherali(SDP) and Sherali(LP) respectively. In the transformed QP form, there are usually linear constraints in addition to quadratic ones. The input data for the algorithms ....

M. V. Ramana, (1993) An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, MD.

....symmetric matrices onto the n dimensional Euclidean space R n . Thirdly, and also most importantly, we can apply the SDP relaxation, which was originally developed for 0 1 integer programming problems by Lov asz and Schrijver [12] and later extended to nonconvex quadratic optimization problems [6, 18, 19], to the entire class of maximization problems having a linear objective function and finitely or infinitely many quadratic inequality constraints. See also [1, 8, 9, 15, 23, 24, 29, etc. In addition to the reasons above, we should mention that the maximization problem with a linear objective ....

M. V. Ramana, An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

....then EDMCP can be solved more simply as two or more smaller subproblems; one for each connected component of G. Thus assume that G is connected. Since G is also assumed to be incomplete, X is not a singleton set. Next we study the facial structure of Omega Gamma (See also the theses [17, 18] and the papers [19, 16, 14] for characterizations for general sets) Definition 3.1 A matrix X 2 Omega is said to be an extreme point if X can not be represented as a proper convex combination of two distinct points X 1 and X 2 in Omega . Lemma 3.1 Let X, Z 2 S n Gamma1 and let X be ....

M.V. RAMANA. An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

....It must be underlined that the idea of formulating non convex control problems as an LMI problem with a rank constraint is not new, see for instance [11] Beyond the scope of systems control, rank constrained LMI problems also occur in mathematical programming and combinatorial optimization. In [28, 26, 8] it is shown that all of the problems with polynomial objective and polynomial constraints (including quadratic programming, binary programming, integer programming and job shop scheduling, to mention just a few) may actually be formulated as rank one LMI optimization problems. 4 Numerical ....

....of the complex plane are also amenable to this unifying treatment. It should also be insightful to assess the applicability to our problem of the various rank minimization heuristics proposed in the literature. For instance, the cutting plane technique for rank one linear programming proposed in [28] may prove useful. Recent theoretical results on the rank minimization problem [25] also deserve to be considered. Finally, it should be highly instructive to study deterministic global optimization techniques [13, 22, 4, 2] that can take advantage of the special structure of the rank one LMI ....

M. V. Ramana, "An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems", Ph. D. Thesis, The Johns Hopkins University, Baltimore, Maryland, 1993.

....in optimization is the General Quadratic Programming Problem (GQP) with quadratic (possibly nonconvex) constraints and objective function. This is a very general NP hard problem [18] including, for example, integer programming and optimization with general polynomial constraints (see, for example [23]) Since the problem is NP hard, many relaxations of the problem have been studied. One convex relaxation of the problem is the semi definite relaxation proposed by Shor [26] see also [30] Much progress was made in the last years in developing polynomialtime interior point methods for ....

M.V. Ramana. An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems. PhD thesis, The Johns Hopkins University, Baltimore, 1993.

....that a polynomial bound has not been established for the bitlengths of the intermediate numbers occurring in these algorithms. ffl For any fixed m, there is a polynomial time algorithm (in n; L) that checks whether there exists an x such that Q(x) 0, and if so, computes such a vector ([Ram93]) In a recent work[Fre95] Bob Freund discusses interior point algorithms for SDPs in which no regularity (Slater like) conditions are assumed. Progression from weak optimization to strong optimization is made difficult as there exist SDPs with certain undesirable features as described in the ....

....a guess vector x, whether Q(x) 0, then it follows that SDFP is in NP. We can 23 perform this task in O(maxfn 3 ; mn 2 g) arithmetic computations as follows: first compute the combination matrix Q(x) in O(mn 2 ) steps and then apply partial Cholesky decomposition (see [GLS88] or [Ram93]) to check if this matrix is positive semidefinite in O(n 3 ) steps. 2 The reason one cannot extend the above proof that SDFP2 NP Co NP for the BSS model to the TM model is because feasible semidefinite inequalities need not always have rational solutions of polynomial size (see examples in ....

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M.V. Ramana,An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, Ph.D. Thesis, The Johns Hopkins University, Baltimore, 1993.

....for solving SDP approximately (see x4.1) In their seminal work, Nesterov and Nemirovskii [NN94] developed efficient interior point methods for a wider class of convex programs, by employing self concordant barrier functions. Other early papers in the area include [Jar91] and [Ove90] In [Ram93], the relationship between SDP and multiquadratic programming (quadratic programming with quadratic constraints) was studied and certain geometric and algorithmic results were developed for SDP. A natural generalization of the standard LP duality has been considered in [Ali95] and [NN94] This can ....

....that a polynomial bound has not been established for the bitlengths of the intermediate numbers occurring in these algorithms. ffl For any fixed m, there is a polynomial time algorithm (in n; L) that checks whether there exists an x such that Q(x) 0, and if so, computes such a vector ([Ram93]) For the nonstrict case of Q(x) 0, the feasibility can be verified in polynomial time for the fixed dimensional problem as shown in [PK95] In a recent work[Fre95] Rob Freund discusses interior point algorithms for SDPs in which no regularity (Slater like) conditions are assumed. 4.2 ....

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M.V. Ramana, An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, Ph.D. Thesis, The Johns Hopkins University, Baltimore, 1993.

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M.V. RAMANA. An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

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M. Ramana. An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems. PhD thesis, The John Hopkins University, Baltimore, 1993.

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M. Ramana. An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems. PhD thesis, The Johns Hopkins University, 1993.

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M.V. RAMANA. An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems. PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

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Ramana, M. V. (1993): An algorithmic analysis of multiquadratic and semidefinite programming problems. PhD thesis, Johns Hopkins University, Baltimore, MD

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Ramana, M. V. (1993): An algorithmic analysis of multiquadratic and semidenite programming problems. PhD thesis, Johns Hopkins University, Baltimore, MD

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M. V. Ramana, An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, Md, 1993.

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Ramana, M. V. (1993): An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, MD.

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M. V. Ramana, An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, MD, 1993.

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