R. Fletcher, Semi-definite matrix constraints in optimization, SIAM J. Control Optim., 23(1985), 493-513.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the maximum eigenvalue of a symmetric matrix that depends affinely on x. In this case, the problem is in fact convex (but still nondifferentiable) Many researchers have considered this problem. Relevant work includes Cullum et al. [CDW75] Craven and Mond [CM81] Polak and Wardi [PW82] Fletcher [Fle85], Shapiro [Sha85] Friedland et al. FNO87] Goh and Teo [GT88] Panier [Pan89] Allwright [All89] Overton [Ove88, Ove92, OW93, OW92] Ringertz [Rin91] Fan and Nekooie [FN92] and Fan [Fan92] In [BY89] Boyd and Yang use the cutting plane algorithm and Shor s subgradient method [Sho85] to solve ....

R. Fletcher. Semidefinite matrix constraints in optimization. SIAM J. Control and Opt., 23:493--513, 1985.

....[24] investigated in detail its application to combinatorial optimization problems by using it to approximate the solution of both LP and SDP relaxations. Lovasz and Schrijver [36] later showed how SDP problems can provide tighter relaxations of (0, 1) programming problems than can LP. Fletcher [17, 18] revived interest in SDP among nonlinear programmers in the 80s, and this led to a series of papers by Overton and Overton and Womersley; see [50] and the references therein. The key contributions of Nesterov and Nemirovski [44, 45] and Alizadeh [1] showed that the new generation of ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM J. Control Optim., 23:493--513, 1985.

....cone and W ae H n be a compact set. Then exactly one of the following holds: a) There is a matrix A 2 K such that tr AB 0 for all B 2 W . b) There are i 0 and B i 2 W such that k X i=1 i B i 2 K : Optimality conditions similar to those in the next lemma are given in [12] and [3]. Lemma 1.4. Let S be a closed convex subset of H n , and let A 2 H n and X 0 2 S be given. Let K be the cone of feasible directions at X 0 . Then the following are equivalent: a) min (A X 0 ) min (A X) for all X 2 S. 4 roy mathias (b) There is a nonzero positive semidefinite matrix M 2 ....

R. Fletcher. Semidefinite matrix constraints in optimization. SIAM J. Control and Opt., 23(4), 1985.

....L n an elliptope (coming from ellipsoid and polytope) The elliptope L n is the central object studied in this paper. By definition, e L n is nothing but a section of the cone PSD n by the hyperplanes x ii = 1 for all i. The cone PSD n has been extensively studied in the literature; see e.g.[2, 8, 15] for results on its faces. As a matter of fact, e L n inherits some of the good properties of PSD n but, however, its structure is much more complicated than that of PSD n . For instance, the description of the faces of e L n follows from that of the faces of PSD n (see Proposition 2.6) but, ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM Journal on Control and Optimization 23 (4) (1985) 493--513.

....yet powerful unifying framework in which to study a wide variety of important results. Examples include Schur convexity (see for example [22] the convexity of eigenvalue functions ( 10, 6, 11, 3, 13, 19] calculations of Fenchel conjugates and subdifferentials of convex eigenvalue functions [24, 5, 12, 30, 28, 25, 26, 27, 15, 16, 1, 17, 19], von Neumann s original result [33] and generalizations (for example [4, 20] subdifferentials of unitarily invariant norms [34, 35, 36, 37, 38, 8, 7, 9, 20] and characterizations of extreme, exposed and smooth points of unit balls [2, 37, 38, 8, 7, 9, 20] This paper concentrates on convexity ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM Journal on Control and Optimization, 23:493--513, 1985.

....applied to reduce the time or space complexity of the only known polynomialtime algorithm for solving MAXCLIQUE and MAX STABLE SET on a perfect graph if its chromatic number is a constant. 8.2 Future Work We propose the following future work. ffl There are plenty of other semidefinite programs [106, 108, 63, 55, 51, 52, 150, 113, 32]. It would be nice if our algorithms could be generalized to work on some of them. ffl Our compact algorithms are still far from practical due to their time complexity. It would be nice if one can come up with compact algorithms with better running times. We propose to perform the conjugate ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM Journal on Control and Optimization, 23(4):493--513, 1985.

....the duality theory from the point of view of basic feasible solutions and extend the tableau based proofs of LP duality. The latest version of Nesterov and Nemirovskii s text [NN92] also treats cone duality for the general convex programs. Papers of Overton and Womersley [OW92] and Fletcher [Fle85] treat duality theory for the eigenvalue optimization problem from the subdifferential point of view. Such an approach is related to the KuhnTucker duality theory and relies on derivatives or subgradients. Also Lov asz in [Lov79] Grotschel, Lov asz and Schrijver [GLS81, GLS84, GLS88] and Shapiro ....

.... V i Gamma (m i Gamma m i 1 )X i )Y i = z i I U i (m i Gamma m i 1 )X i )W i = I Gamma Y i )U i = I Gamma W i )V i = 0 for i = 1; Delta Delta Delta ; k. The characterization (4.40) and the max part of (4. 50) were given in Overton and Womersley [OW91] Also, Fletcher in [Fle85] derives a closely related result to (4.40) but the result was incorrect (Fletcher had 0 S rather than 0 S I. The min characterizations as well as the primal and dual formulation of the variants with equality constraints, we believe are new. Similar formulations can be derived for maximizing ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM J. Control Optim., 23:493--513, 1985.

....applications many other types arise (e.g. Toeplitz structure) 1. 2 Historical overview An early paper on the theoretical properties of semidefinite programs is Bellman and Fan [BF63] Other references discussing optimality conditions are Craven and Mond [CM81] Shapiro [Sha85] Fletcher [Fle85], Allwright [All88] Wolkowicz [Wol81] and Kojima, Kojima and Hara [KKH94] Many researchers have worked on the problem of minimizing the maximum eigenvalue of a symmetric matrix, which can be cast as a semidefinite program (see x2) See, for instance, Cullum, Donath and Wolfe [CDW75] Goh and ....

....containing all the x i (shown as stars) and none of the y j (shown as circles) Finding such an ellipsoid can be cast as a semidefinite program, hence efficiently solved. 10 Statistics Semidefinite programs arise in minimum trace factor analysis (see Bentler and Woodward [BW80] Fletcher [Fle81, Fle85], Shapiro [Sha82] Watson [Wat92a] Assume x 2 R p is a random vector, with mean x and covariance matrix Sigma. We take a large number of samples y = x n, where the measurement noise n has zero mean, is uncorrelated with x, and has an unknown but diagonal covariance matrix D. It follows ....

R. Fletcher. Semidefinite matrix constraints in optimization. SIAM J. Control and Opt., 23:493--513, 1985.

.... The efficiency of recent interior point methods for SDP, which is directly responsible for the popularity of SDP in control, has therefore also attracted a great deal of interest in optimization circles, overshadowing earlier solution methods based on techniques from nondifferentiable optimization [8, 10, 11, 12, 13]. At every major optimization conference, there are workshops and special sessions devoted exclusively to SDP, and a special issue of Mathematical Programming has recently been devoted to SDP [14] This interest was primarily motivated by applications of SDP in combinatorial optimization but, more ....

R. Fletcher, "Semidefinite matrix constraints in optimization", SIAM J. Control and Opt., vol. 23, pp. 493--513, 1985.

.... nonlinear program: minimize f(x) subject to g(x) # 0, where f : # n ## # and g = g ij ] m i,j=1 : # n ## S are twice continuously di#erentiable functions and, for A # S, A # 0 (respectively, A # 0 ) means A is positive semidefinite (respectively, positive definite) [2, 13, 19, 24, 26]. Here W denotes the space of mm blockdiagonal real matrices with k blocks of sizes m 1 , m k , respectively (the blocks are fixed) and S denotes the subspace comprising those A # W that are symmetric, i.e. A T = A. We sketch the extension below. We endow W with the inner product ....

Fletcher, R., Semi-definite matrix constraints in optimization, SIAM J. Control Optim., 23 (1985), 493-513.

....of nonlinear and time varying systems, and for controller synthesis. This is of great interest to computeraided control system design. SDPs also arise in other fields of engineering and mathematics, such as structural optimization, circuit design, signal processing, and statistics, see e.g. [11, 62, 63, 17, 18, 50, 56]. For other applications include solving Ricatti equations, see e.g. 13, 64] In this arena SDP is usually known as LMI: Linear Matrix Inequalities. A typical example of an SDP application in Systems and Control is the numerical search for Lyapunov inequalities and also for Lyapunov functions ....

R. FLETCHER. Semi-definite matrix constraints in optimization. SIAM J. Control and Optimization, 23:493--513, 1985.

....Chap. XV] which are very efficient to solve large scale problems with a moderate accuracy. When high accuracy is needed, second order information must be added in the model. Combining a geometrical and the Sequential Quadratic Programming approaches, a local algorithm was presented and analyzed in [10], 34] 36] 35] and [41] in the latter two papers, a quadratic rate of convergence was obtained. Yet, in this SQP framework, the authors considered only a local analysis; issues of global convergence were not addressed. In this paper, we present, as in [33] the second order analysis of the ....

....semidefinite, a consequence of the Schur product Theorem [17, x 5.2] is that tr XZ = 0 if and only if XZ = 0. Then, via (i) Z has the form QY Q T ; it is positive semidefinite if and only if Y 0. 2 The following theorem recalls previously known geometrical descriptions of 1 (X) see [10] or [34] and, along the lines of [15] makes an explicit link with the exposed faces of C n . Theorem 2.3 (i) Let X 2 S n and let Q 1 be an n Theta r matrix whose columns form an orthonormal basis of E 1 (X) The face of C n exposed by X is F Cn (X) fQ 1 Y Q T 1 : Y 2 C r g = cofqq T : ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM J. Control Optim., 23:493--523, 1985.

....k may not be differentiable. Since we want X orthogonal at the solution, we can expect multiple eigenvalues of 1. It is well known that the largest eigenvalue is convex and so we can obtain expressions for the subdifferentials of the largest eigenvalue if its multiplicity is 1, see e.g. [13], 23] These subdifferentials are just the convex hull, of the expression given above for the derivative, over all normalized eigenvectors v k . Note that the differentials at X in the direction h of the above functions in (P ) are: df(X; h) trA(XBh t hB t X t ) 2Ch t dg(X; h) Xh ....

R. FLETCHER. Semi-definite matrix constraints in optimization. SIAM J. Control and Optimization, 23:493--513, 1985.

....on the set of s.p.d. matrices and thus any stationary point is a global minimizer. Proof. The result 1. without the function oe , is given in [9, Prop 2.1] Including oe follows from the definitions, as does 2. On the set of s.p.d. matrices, the largest eigenvalue is a convex function, see e.g. [16, 17], while det(B) 1 n is an increasing concave function, see e.g. 16, pg 475] By increasing we mean isotonic with the Loewner order, i.e. the order A B if A Gamma B is s.p.d. Thus oe is pseudoconvex, see e.g. 18] For our purposes we need to only know that the ratio of a convex and ....

....function in the case of a multiple eigenvalue. However, since it is a real valued convex function, it is continuous and subdifferentiable, see e.g. 19] The subgradient consists of the convex hull of the normalized m eigenvectors, where m is the multiplicity of the eigenvalue 1 , see e.g. [16, 17]. We can now find a subgradient of the Lagrangian, using the calculations in the proof of Theorem 5.1 in [9] and set it equal to zero. Using the fact that the derivative of the determinant is the adjoint matrix and that Cramer s rule states that the inverse is the adjoint divided by the ....

R. FLETCHER. Semi-definite matrix constraints in optimization. SIAM J. Control and Optimization, 23:493--513, 1985.

....[ Gamma1; 1] is the lower semicontinuous, convex function f (y) supfx T y Gamma f(x) j x 2 R n g: We will make frequent use of ideas and notation from [26] By analogy, for a matrix function F : H ( Gamma1; 1] we can define a conjugate matrix function F : H [ Gamma1; 1] c.f. [8]) by F (Y ) supftrXY Gamma F (X) j X 2 Hg: 1.2) Exactly as in R n , because F is expressed as a supremum of (continuous) linear functions of Y , it must be convex and lower semicontinuous. The idea of our key result is then rather simple. We will prove (Theorem 2.6) that if the ....

....setting analogous to those in our present development: such results for unitarily invariant norms have appeared in [3, 36] Studying convex matrix functions via their Fenchel conjugates is not a new idea. It is implicit for example in some of the techniques in [7] and was used explicitly in [8] to study the sum of the largest k eigenvalues of a real symmetric matrix, an approach also followed in [12] see also [13] The primary aim of these latter papers is to study sensitivity results via the subdifferential set. Various representations of this set were investigated in [22, 23, 24] ....

[Article contains additional citation context not shown here]

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM Journal on Control and Optimization, 23:493--513, 1985.

....connection with the approximation of the max cut problem. Some related topics. By definition, the convex set L n is nothing but a section of the cone PSD n formed with the intersection by the hyperplanes x ii = 1 for all i. The cone PSD n has been extensively studied in the literature; see e.g.[2, 9, 16] for results on its faces. As a matter of fact, L n inherits some of the good properties of PSD n . However, its structure is much more complicated than that of PSD n . For instance, the description of the faces of L n follows from that of the faces of PSD n (see Proposition 2.7) But, while each ....

R. Fletcher. Semi-definite matrix constraints in optimization. SIAM Journal on Control and Optimization 23 (4) (1985) 493--513.

....efficient region updates to choose; 2. improving the efficient region by finding a properly efficient region; 90 3. trying to have some convergence analysis for the updates in the efficient region. Appendix A Subgradients for Largest and Smallest Eigenvalues We include the results in [37] and [38] and also see reference [39] and [40] Let X be the space of n Theta n symmetric matrices A. Define the inner product for X by trace (A t B) where A; B 2 X : Denote B 0 if B 2 X is positive semi definite. Consider the function 1 (A) which is the largest eigenvalues of A 2 X . For the ....

R. FLETCHER. Semi-definite matrix constraints in optimization. SIAM J. Control and Optimization, 23:493--513, 1985.

....in a certain fixed region, say the right half of the complex plane. We shall refer readers to papers [4, 16, 26, 85, 90] for Problem AD and Problem MD, and not give any more reviews here. Related to Problem AD is the communality problem in factor analysis [83] and the educational testing problem [23, 56]. The former deals with finding a diagonal matrix X so that the matrix A X in which A is a given real symmetric matrix with zero diagonal entries has many eigenvalues equal to zero as possible. For other related work of the communality problem see [103] The latter concerns finding a ....

R. FLETCHER, Semi-definite matrix constraints in optimization, SIAM J. Control Optim., 23 (1985), pp. 493-513.

....theory from the point of view of basic feasible solutions 6 F. Alizadeh and extend the tableau based proofs of LP duality. The latest version of Nesterov and Nemirovskii s text [50] also treats cone duality for the general convex programs. Papers of Overton and Womersley [53] and Fletcher [23] treat duality theory for the eigenvalue optimization problem from the point of view of subdifferentials. Such an approach is related to the Kuhn Tucker duality theory and relies on derivatives or subgradients. Also Lov asz in [41] Grotschel, Lov asz and Schrijver [30, 31, 32] and Shapiro in ....

....i Gammam i 1 )X i )Y i = z i I U i (m i Gammam i 1 )X i )W i = I GammaY i )U i = I GammaW i )V i = 0 for i = 1; Delta Delta Delta ; k. The characterization (4.3) and the max part of (4. 13) were given in Overton and Womersley [54] Also, Fletcher in [23] derives a closely related result to (4.3) but the result was incorrect (Fletcher had 0 S rather than 0 S I. The min characterizations as well as the primal and dual formulation of the variants with equality constraints, we believe are new. In a similar manner, primal and dual SDP ....

R. Fletcher, Semi-definite matrix constraints in optimization, SIAM J. Control Optim., 23 (1985), pp. 493--513.

No context found.

R. Fletcher, Semi-definite matrix constraints in optimization, SIAM J. Control Optim., 23(1985), 493-513.

No context found.

R. Fletcher. Semidefinite matrix constraints in optimization. SIAM J. Control and Opt., 23: 493--513, 1985.

No context found.

R. Fletcher. Semidefinite matrix constraints in optimization. SIAM J. Control and Opt., 23:493--513, 1985.

No context found.

R. FLETCHER. Semi-definite matrix constraints in optimization. SIAM J. Control Optim., 23:493-- 513, 1985.

No context found.

R. Fletcher [1985], Semi-definite matrix constraints in optimization, SIAM Journal on Control and Optimization 23, pp. 493--513.

No context found.

R. Fletcher (1985), "Semi-definite matrix constraints in optimization", SIAM Journal on Control and Optimization 23, pp. 493-513.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC