A CONIC DANTZIG-WOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING
|
|
- Alice McDonald
- 6 years ago
- Views:
Transcription
1 A CONIC DANTZIG-WOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING Kartik Krishnan Advanced Optimization Laboratory McMaster University Joint work with Gema Plaza Martinez and Tamás Terlaky Computational Mathematics Seminar University of Waterloo January 24, 2005
2 Contents ˆ Conic programming ˆ Motivation and Previous work ˆ Semidefinite programming ˆ Conic Dantzig-Wolfe decomposition ˆ Algorithm ˆ Implementational Issues in Algorithm ˆ Computational results ˆ Conclusions and Future work -1-
3 Conic programming (P) min c T x s.t. Ax = b x K (D) max b T y s.t. A T y + s = c s K where A R m n, b R m, c R n, K = K 1... K r ˆ r = 1, K = R n + = {x R n : x 0} LP Very large LPs (m, n 1,000,000) solvable by the simplex method and/or IPMs. ˆ K i = Q n i + = {x R n i : x 1 x 2:ni } SOCP Large SOCPs (m, n 100,000) solvable by IPMs. ˆ K i = S n i + = {X S n i : X 0} SDP Medium sized SDPs (m, n 1000) solvable by IPMs. (Beyond 10,000 seems impossible today!) -2-
4 Applications of conic programming 1. Combinatorial optimization: Alizadeh, Goemans-Williamson, Lovasz-Schrijver, Ye. 2. Linear control theory: Boyd, Vandenberghe, Balakrishnan, El Ghaoui, Feron. 3. Signal Processing: Luo. 4. Robust optimization: Ben Tal-Nemirovski, El Ghaoui. 5. Polynomial programming: Bertsimas, Lasserre, Nesterov, Parrilo. 6. Others include finance, statistics, moment problems etc. -3-
5 Motivation (a) Solve large scale semidefinite programs (SDP) arising in science and engineering. (b) Interior Point Methods (IPM) can solve large scale linear programming (LP) and second order cone programming (SOCP) problems. However, they are limited in the size of SDPs they can solve. (c) The technique is to decompose an SDP into mixed conic programs with LP, SOCP and smaller SDP blocks. (Conic Dantzig-Wolfe decomposition scheme) -4-
6 Survey of approaches Dantzig-Wolfe (1960) Decomposition approach for Kelley (1960) large scale LPs. Karmarkar et al. ( ) Interior point methods. ACCPM (Goffin, Vial et al.) Interior point cutting plane LBCPM (Roos et al., methods. Gondzio, Mitchell) Spectral bundle (Helmberg et al.) Lagrangian relaxation Volume algorithm (Barahona et al.) approaches Krishnan-Mitchell (2001) LP decomposition approach for Goldfarb (2002) semidefinite/conic programming. Our present work, Extend this decomposition Oskoorouchi-Goffin (2004) approach to a mixed conic setting (LP, SOCP, smaller SDP blocks). -5-
7 Semidefinite programming (SDP) min C X s.t. A(X) = b X 0 (SDD) max b T y s.t. A T y + S = C S 0 Notation - X, S, C S n, b R m - A B = trace(ab) = n A ij B ij (Frobenius inner product) i,j=1 - The operator A : S n R m and its adjoint A T : R m S n are A(X) = A 1 X m., A y = y i A i A m X i=1 where A i S n, i = 1,..., m -6- First Prev Next Last Go Back Full Screen Close Quit
8 Assumptions (1) A 1,..., A m are linearly independent in S n. (2) Slater condition for (SDP) and (SDD): {X S n : A(X) = b, X 0} {(y, S) R m S n : A T y + S = C, S 0} (3) One of the constraints in (SDP) is I X = 1 (trace(x) = 1). (Every SDP with a bounded primal feasible set can be reformulated to satisfy this assumption) -7-
9 Computational effort in each iteration 1. SDP: ˆ Form the Schur matrix M of size m, whose (i, j)th entry is given by M ij = A j XA i S 1. This requires O(mn 3 + m 2 n 2 ) flops, and is the costliest step in all. ˆ The matrix M is dense despite the sparsity in A i and factoring it requires O(m 3 ) flops. ˆ The line search involves a Cholesky factorization of X and S which is O(n 3 ) flops. 2. LP and SOCP: The costliest step is in factorizing M. ˆ In LP, M inherits the sparsity in A. ˆ In SOCP, if A is sparse, M can be expressed as a sparse matrix plus some low rank updates. -8- First Prev Next Last Go Back Full Screen Close Quit
10 Conic Dantzig-Wolfe decomposition ˆ Our semidefinite programs are: ˆ Consider the set: ˆ The extreme points of E are: (SDP) min C X s.t. A(X) = b trace(x) = 1 X 0 (SDD) max b T y + z s.t. A T y + zi + S = C S 0 E = {X S n : trace(x) = 1, X 0} {vv T : v R n, v T v = 1} (Infinite number) -9-
11 Conic Dantzig-Wolfe decomposition: Primal motivation Any X E can be written as: (1) X = j λ j d j d T j where j λ j = 1, λ j 0 X 0 is replaced by λ j 0, j (Semi-infinite LP formulation) (2) X = j P j V j P T j where j trace(v j ) = 1, V j S 2 + X 0 is replaced by V j 0, j (Semi-infinite SOCP formulation) (A 2 2 SDP constraint can be written as a size 3 SOCP constraint) (3) X = j P j V j P T j where j trace(v j ) = 1, V j S r j +, r j 3 X 0 is replaced by V j 0, j (Semi-infinite SDP formulation) -10- First Prev Next Last Go Back Full Screen Close Quit
12 A 2 2 SDP constraint is an SOCP constraint of size 3 X = ( ) X11 X 12 X 12 X 22 0 X 11 0, X 22 0, X 11 X 22 X X 11 + X 22 2X 12 X 11 X22 Q First Prev Next Last Go Back Full Screen Close Quit
13 Conic Dantzig-Wolfe decomposition: Dual motivation (1) LP formulation: S 0 d T j Sd j 0, j (if we take a finite number of constraints we obtain an LP relaxation) (2) SOCP formulation: S 0 P T j SP j }{{} 2 2 0, j (finite number of constraints gives SOCP relaxation) (3) SDP formulation: S 0 P T j SP j }{{} 0, j (r j 3, j) r j r j (finite number of constraints gives SDP relaxation) -11-
14 Conic Dantzig-Wolfe decomposition: Master problem 1 The decomposed conic problem over LP, SOCP and SDP blocks is: Primal Dual min s.t. n l i=1 n l n q n s c li x li + c T qjx qj + C sk X sk j=1 k=1 A li x li + A qj x qj + A sk (X sk ) = b i=1 n l n q j=1 n s k=1 x li + x qj1 + trace(x sk ) = 1 i=1 n q j=1 n s k=1 x li 0, i = 1,..., n l x qj Q 0, j = 1,..., n q X sk 0, k = 1,..., n s max b T y + z s.t. A T li y + z + s li = 1 c li, i = 1,..., n l A T qj y + z 0 + s qj 0 = c qj, j = 1,..., n q A T sk y + zi r k + S sk = C sk, k = 1,..., n s s li 0, i = 1,..., n l s qj Q 0, j = 1,..., n q S sk 0, k = 1,..., n s -12- First Prev Next Last Go Back Full Screen Close Quit
15 Conic Dantzig-Wolfe decomposition: Master problem 2 1. The dual master problem is a relaxation of (SDD). 2. The primal master problem is a constrained version of (SDP), i.e., any feasible solution in the primal master problem is feasible in (SDP). 3. IPMs are well equipped to handle mixed conic problems over LP, SOCP, and SDP cones. 4. We will restrict the size of the SDP blocks to about r = O( n) in the master problem. -13-
16 Conic Dantzig-Wolfe decomposition: Separation Oracle Input: (y, z ) a feasible point for dual master problem. ˆ If λ min (S ) 0, report feasibility STOP. ˆ Else, solve the following subproblem for X : min (C A T y z I) X s.t. I X = 1 X 0 Factorize X = DMD T with M 0. Return the cut D T (C A T y zi)d 0 with D R n r. If r = 1 is an LP cut. If r = 2 is an SOCP cut. If r 3 is an SDP cut of small size. -14-
17 Conic Dantzig-Wolfe Algorithm -15-
18 Choice of the query point: I ˆ In the original Dantzig-Wolfe (Kelley) scheme the query point (y, z) is an optimal solution to the dual master problem. This scheme has a poor rate of convergence. ˆ Better to solve the master problem approximately initially, and gradually tighten this tolerance as the algorithm proceeds. Initially, the master problem is a poor approximation to the SDP problem. Weak tolerance Central dual iterates (y, z) Oracle returns better cuts As the algorithm proceeds, the master problem approximations get tighter. Tight tolerance More emphasis on the objective function Convergence to an optimal solution -16-
19 Choice of the query point : 2 We adopt the following adaptive strategy for a prescribed tolerance TOL in every iteration: ˆ Solve the master problem to a tolerance TOL to obtain the solution (x, y, z, s ). Compute the following parameters: GAPTOL(x, s ) = x T s max{1, 1 2 (ct x +b T y )} λ min (C A T y z I) max{1, C } INF(y, z ) = OPT(x, y, z, s ) = max{gaptol(x, s ), INF(y, z )} ˆ If OPT(x, y, z, s ) < TOL, we lower TOL by a constant factor. More precisely, TOL = µ OPT(x, y, z, s ) with 0 < µ < 1. Else, TOL remains unchanged. -17-
20 Algorithm -18-
21 One iteration of Algorithm (y,z) space y New feasible region (SDD) feasible region A D B z New (y,z) Cutting plane C A Central path Old (y,z) Feasible region of dual master problem A : Solve master problem to tolerance TOL B : Oracle returns a cutting plane C : Restoring dual feasibility D : Warm start -19- First Prev Next Last Go Back Full Screen Close Quit
22 Upper and lower bounds ˆ The objective value of the primal master problem in every iteration gives an upper bound on the SDP objective value. This is the objective value of the dual master problem plus the duality gap. ˆ Given a solution (y, z ) to the dual master problem in every iteration, a lower bound is computed as follows: Compute λ = λ min (S ), where S = (C A T y z I). Set y lb = y and z lb = z + λ. A lower bound on the SDP objective value is then b t y lb + z lb Note: (y lb, z lb ) is a feasible point in (SDD). ˆ We could also terminate if the difference between these bounds is small. ˆ The computed lower (upper) bounds do not increase (decrease) monotonically. -20-
23 Warm start after adding a column: 1 ˆ The solution to the old master problem is (x l, x q, y, s l, s q). Consider the linear cut a T l y d in the dual master problem. The new master problem is : min c T l x l + c T q x q + d T β s.t. A l x l + A q x q + a l β = b x l 0, β 0 x q Q 0. max b T y s.t. s l = c l A T l y 0 s q = c q A T q y Q 0 γ = d a T l y 0 ˆ We have γ < 0. We can perturb y (lower bounding) to generate y lb so that γ lb = 0. ˆ The perturbed point (x l, x q, β, y lb, s lb l, s lb q, γ lb ) is feasible in the new master problem with β = γ lb = 0, but NOT STRICTLY!. ˆ We want to increase β and γ lb from their current zero values, while limiting the variation in the other variables. -21-
24 Warm start after adding a column: 2 ˆ We solve the following problems (WSP) max log β s.t. A l x l + A q x q + a l β = 0 D 1 l β 0. x l 2 + Dq 1 x q 2 1 (WSD) max log γ s.t. a T l y + γ = 0 D l A T l y 2 + D q A T q y 2 1 γ 0. for ( x l, x q, β) and ( y, γ) respectively. Here D l and D q are appropriate primal-dual scaling matrices for LP and SOCP blocks at (x l, x q, s lb l, s lb q ). ˆ Compute ( s l, s q ) = ( A T l y, A T q y) First Prev Next Last Go Back Full Screen Close Quit
25 Warm start after adding a column: 3 ˆ The solutions to (WSP) and (WSD) are given by x l = Dl 2 A T l (A l Dl 2 A T l + A q DqA 2 T q ) 1 a l β x q = DqA 2 T q (A l Dl 2 A T l + A q DqA 2 T q ) 1 a l β y = (A l Dl 2 A T l + A q DqA 2 T q ) 1 a l β where β R is the solution to max{ 1 2 βt V β + log β : β 0} where V = a T l (A l D 2 l A T l + A q D 2 qa T q ) 1 a l and γ = V β. ˆ Finally set (x st l, x st q, β st ) = (x l αmax x p l, x q αmax x p q, αmaxβ) p y st = yl lb αmax y d αmax s d l, s lb q αmax s d q, αmaxγ) d (s st l, s st q, γ st ) = (s lb l where α p max and α d max are the maximal primal and dual step lengths respectively First Prev Next Last Go Back Full Screen Close Quit
26 Warm start after adding an SOCP cut: 1 ˆ The solution to the old master problem is (x l, x q, y, s l, s q). Consider the SOCP cut a T q y Q d in the dual master problem. The new master problem is : min c T l x l + c T q x q + d T β s.t. A l x l + A q x q + a q β = b x l 0, β Q 0 x q Q 0. max b T y s.t. s l = c l A T l y 0 s q = c q A T q y Q 0 γ = d a T q y Q 0 ˆ We have γ Q 0. We can perturb y (lower bounding) to generate y lb so that γ lb Q 0. ˆ The perturbed point (x l, x q, β, y lb, s lb l, s lb q, γ lb ) is feasible in the new master problem with β = 0 and γ lb Q 0, but NOT STRICTLY!. ˆ We want to increase β and γ lb, making them strictly feasible, while limiting the variation in the other variables. -24-
27 Warm start after adding an SOCP matrix: 2 ˆ We solve the following problems (WSP) max 1 2 log(β2 1 β2 2 β3) 2 s.t. A l x l + A q x q + a q β = 0 D 1 l x l 2 + Dq 1 x q 2 1 β Q 0. (WSD) max 1 2 log(γ2 1 γ 2 2 γ 2 3) s.t. a T q y + γ = 0 D l A T l y 2 + D q A T q y 2 1 γ Q 0. for ( x l, x q, β) and ( y, γ) respectively. Here D l and D q are appropriate primal-dual scaling matrices for LP and SOCP blocks at (x l, x q, s lb l, s lb q ). ˆ Compute ( s l, s q ) = ( A T l y, A T q y) First Prev Next Last Go Back Full Screen Close Quit
28 Warm start after adding an SOCP matrix: 3 ˆ Compute x l = Dl 2 A T l (A l Dl 2 A T l + A q DqA 2 T q ) 1 a q β x q = DqA 2 T q (A l Dl 2 A T l + A q DqA 2 T q ) 1 a q β y = (A l Dl 2 A T l + A q DqA 2 T q ) 1 a q β where β R 3 is the solution to max{ 1 2 βt V β log(β2 1 β 2 2 β 2 3) : β Q 0} where V = a T q (A l D 2 l A T l + A q D 2 qa T q ) 1 a q and γ = V β. ˆ Finally set (x st l, x st q, β st ) = (x l αmax x p l, x q αmax x p q, αmaxβ) p y st = yl lb αmax y d αmax s d l, s lb q αmax s d q, αmaxγ) d (s st l, s st q, γ st ) = (s lb l where α p max and α d max are the maximal primal and dual step lengths respectively First Prev Next Last Go Back Full Screen Close Quit
29 Computational Results ˆ Matlab code with the following features: Currently solves SDP over LP and SOCP blocks. SDPT3 is used as conic solver for master problem. Oracle is implemented using MATLAB s Lanczos solver (eigs). Warm-starting and lower-upper bounding procedures built in. ˆ Tested it on Max-Cut problems (SDPLIB, DIMACS set). -27-
30 Max-Cut problem ˆ Primal min L X s.t. X ii = 1, i = 1,..., n X 0 ˆ Dual max e T y s.t. S = L Diag(y) 0 ˆ n is the number of nodes of the graph. ˆ L is the Laplacian matrix of the graph. ˆ This is an SDP relaxation of the Max-Cut problem. -28-
31 Computational results for Max-Cut: 1 ˆ Our starting dual master problem is max e T y s.t. y i L ii, i = 1,..., n ˆ We solve the initial problem to a tolerance of 1e 3, and set TOL = INF(y). ˆ Whenever OPT(x, y, s) < TOL, we set TOL = 1 2 OPT(x, y, s). ˆ Our stopping criterion is OPT(x, y, s) < 1e 3 or n iterations, whichever comes earlier. ˆ The oracle returns SOCP cuts if the two smallest eigenvalues of S were negative, and LP cuts if only λ min (S) was negative First Prev Next Last Go Back Full Screen Close Quit
32 Computational results for Max-Cut: 2 Problem n Optimal LP SOCP Upper Lower OPT value blocks blocks bound bound (x, y, s) mcp e-3 mcp e-3 mcp e-3 mcp e-3 mcp e-3 mcp e-3 mcp e-3 toruspm e-3 torusg e-3 1. OPT(x, y, s) = max{gaptol(x, s), INF(y)}. 2. The first 7 problems are from the SDPLIB repository, while the last two problems are from the DIMACS Challenge. -30-
33 Objective Value Variation of bounds for the toruspm problem Lower Bounds Upper Bounds Best Lower Bound Best Upper Bound Optimal Objective Value Iteration Number
34 Objective Value Variation of bounds for the torusg3 8 problem Lower Bounds Upper Bounds Best Lower Bound Best Upper Bound Optimal Objective Value Iteration Number
35 Future work ˆ Incorporate smaller SDP blocks in master problem. (Already under way) ˆ Procedure to remove unimportant constraints. (To keep the problem small) ˆ Improve and benchmark our decomposition scheme on general SDPs. ˆ Explore the convergence and complexity issues of decomposition scheme. -32-
36 Thank you for your attention!. Questions, Comments, Suggestions? The slides from this talk are available online at kartik/waterloo-math-seminar.pdf -33-
37 Bibliography [1 ] G. B. Dantzig and P. Wolfe, The Decomposition Algorithm for Linear Programming, Operations Res., 8, 1960, pp [2 ] J. E. Kelley, The Cutting Plane Method for solving Convex Programs, J. Soc. Ind. Appl. Math., 8, 1960, pp [3 ] K. Krishnan and J. Mitchell, Properties of a cutting plane algorithm for semidefinite programming, Technical Report, Dept. of Mathematical Sciences, Rensselaer Polytechnic Institute, May 2003 (submitted to the SIAM J. Optim.) [4 ] M. R. Oskoorouchi and J.-L. Goffin, A matrix generation approach for eigenvalue optimization, College of Business Administration, California State University, San Marcos, October [5 ] J.-L. Goffin and J.-P. Vial, Multiple cuts in the analytic center cutting plane method, SIAM Journal on Optimization, 11(2000), pp [6 ] B. Borchers, SDPLIB 1.2, A library of semidefinite programming test problems, Optimization Methods and Software, 11, 1999, pp [7 ] R. H. Tutuncu, K. C. Toh, and M. J. Todd, SDPT3 - a Matlab software package for semidefinite-quadratic-linear programming, version 3.0, August 21, 2001 [8 ] DIMACS 7 th challenge: [9 ] K. Krishnan, G. Plaza, and T. Terlaky, A conic interior point Dantzig-Wolfe decomposition approach for large scale semidefinite programming, forthcoming First Prev Next Last Go Back Full Screen Close Quit
A CONIC INTERIOR POINT DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING
A CONIC INTERIOR POINT DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING Kartik Krishnan Sivaramakrishnan Department of Mathematics NC State University kksivara@ncsu.edu http://www4.ncsu.edu/
More informationA conic interior point decomposition approach for large scale semidefinite programming
A conic interior point decomposition approach for large scale semidefinite programming Kartik Sivaramakrishnan 1 Department of Mathematics North Carolina State University Raleigh, NC 27695-8205 kksivara@ncsu.edu
More informationProperties of a Cutting Plane Method for Semidefinite Programming 1
Properties of a Cutting Plane Method for Semidefinite Programming 1 Kartik Krishnan Sivaramakrishnan Axioma, Inc. Atlanta, GA 30350 kksivara@gmail.com http://www4.ncsu.edu/ kksivara John E. Mitchell Mathematical
More informationAdvances in Convex Optimization: Theory, Algorithms, and Applications
Advances in Convex Optimization: Theory, Algorithms, and Applications Stephen Boyd Electrical Engineering Department Stanford University (joint work with Lieven Vandenberghe, UCLA) ISIT 02 ISIT 02 Lausanne
More informationA PARALLEL INTERIOR POINT DECOMPOSITION ALGORITHM FOR BLOCK-ANGULAR SEMIDEFINITE PROGRAMS
A PARALLEL INTERIOR POINT DECOMPOSITION ALGORITHM FOR BLOCK-ANGULAR SEMIDEFINITE PROGRAMS Kartik K. Sivaramakrishnan Department of Mathematics North Carolina State University kksivara@ncsu.edu http://www4.ncsu.edu/
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More informationA LINEAR PROGRAMMING APPROACH TO SEMIDEFINITE PROGRAMMING PROBLEMS
A LINEAR PROGRAMMING APPROACH TO SEMIDEFINITE PROGRAMMING PROBLEMS KARTIK KRISHNAN AND JOHN E. MITCHELL Abstract. Until recently, the study of interior point methods has doated algorithmic research in
More informationA PARALLEL TWO-STAGE INTERIOR POINT DECOMPOSITION ALGORITHM FOR LARGE SCALE STRUCTURED SEMIDEFINITE PROGRAMS
A PARALLEL TWO-STAGE INTERIOR POINT DECOMPOSITION ALGORITHM FOR LARGE SCALE STRUCTURED SEMIDEFINITE PROGRAMS Kartik K. Sivaramakrishnan Research Division Axioma, Inc., Atlanta kksivara@axiomainc.com http://www4.ncsu.edu/
More informationLecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min.
MA 796S: Convex Optimization and Interior Point Methods October 8, 2007 Lecture 1 Lecturer: Kartik Sivaramakrishnan Scribe: Kartik Sivaramakrishnan 1 Conic programming Consider the conic program min s.t.
More informationInterior-Point versus Simplex methods for Integer Programming Branch-and-Bound
Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Samir Elhedhli elhedhli@uwaterloo.ca Department of Management Sciences, University of Waterloo, Canada Page of 4 McMaster
More informationPrimal-Dual Geometry of Level Sets and their Explanatory Value of the Practical Performance of Interior-Point Methods for Conic Optimization
Primal-Dual Geometry of Level Sets and their Explanatory Value of the Practical Performance of Interior-Point Methods for Conic Optimization Robert M. Freund M.I.T. June, 2010 from papers in SIOPT, Mathematics
More informationResearch Note. A New Infeasible Interior-Point Algorithm with Full Nesterov-Todd Step for Semi-Definite Optimization
Iranian Journal of Operations Research Vol. 4, No. 1, 2013, pp. 88-107 Research Note A New Infeasible Interior-Point Algorithm with Full Nesterov-Todd Step for Semi-Definite Optimization B. Kheirfam We
More informationA PARALLEL INTERIOR POINT DECOMPOSITION ALGORITHM FOR BLOCK-ANGULAR SEMIDEFINITE PROGRAMS IN POLYNOMIAL OPTIMIZATION
A PARALLEL INTERIOR POINT DECOMPOSITION ALGORITHM FOR BLOCK-ANGULAR SEMIDEFINITE PROGRAMS IN POLYNOMIAL OPTIMIZATION Kartik K. Sivaramakrishnan Department of Mathematics North Carolina State University
More informationThe maximal stable set problem : Copositive programming and Semidefinite Relaxations
The maximal stable set problem : Copositive programming and Semidefinite Relaxations Kartik Krishnan Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA kartis@rpi.edu
More informationA semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint
Iranian Journal of Operations Research Vol. 2, No. 2, 20, pp. 29-34 A semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint M. Salahi Semidefinite
More informationarxiv: v1 [math.oc] 26 Sep 2015
arxiv:1509.08021v1 [math.oc] 26 Sep 2015 Degeneracy in Maximal Clique Decomposition for Semidefinite Programs Arvind U. Raghunathan and Andrew V. Knyazev Mitsubishi Electric Research Laboratories 201 Broadway,
More informationProjection methods to solve SDP
Projection methods to solve SDP Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria F. Rendl, Oberwolfach Seminar, May 2010 p.1/32 Overview Augmented Primal-Dual Method
More informationSparse Optimization Lecture: Basic Sparse Optimization Models
Sparse Optimization Lecture: Basic Sparse Optimization Models Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know basic l 1, l 2,1, and nuclear-norm
More informationOn the Sandwich Theorem and a approximation algorithm for MAX CUT
On the Sandwich Theorem and a 0.878-approximation algorithm for MAX CUT Kees Roos Technische Universiteit Delft Faculteit Electrotechniek. Wiskunde en Informatica e-mail: C.Roos@its.tudelft.nl URL: http://ssor.twi.tudelft.nl/
More informationParallel implementation of primal-dual interior-point methods for semidefinite programs
Parallel implementation of primal-dual interior-point methods for semidefinite programs Masakazu Kojima, Kazuhide Nakata Katsuki Fujisawa and Makoto Yamashita 3rd Annual McMaster Optimization Conference:
More informationSemidefinite Programming, Combinatorial Optimization and Real Algebraic Geometry
Semidefinite Programming, Combinatorial Optimization and Real Algebraic Geometry assoc. prof., Ph.D. 1 1 UNM - Faculty of information studies Edinburgh, 16. September 2014 Outline Introduction Definition
More informationw Kluwer Academic Publishers Boston/Dordrecht/London HANDBOOK OF SEMIDEFINITE PROGRAMMING Theory, Algorithms, and Applications
HANDBOOK OF SEMIDEFINITE PROGRAMMING Theory, Algorithms, and Applications Edited by Henry Wolkowicz Department of Combinatorics and Optimization Faculty of Mathematics University of Waterloo Waterloo,
More informationPOLYNOMIAL OPTIMIZATION WITH SUMS-OF-SQUARES INTERPOLANTS
POLYNOMIAL OPTIMIZATION WITH SUMS-OF-SQUARES INTERPOLANTS Sercan Yıldız syildiz@samsi.info in collaboration with Dávid Papp (NCSU) OPT Transition Workshop May 02, 2017 OUTLINE Polynomial optimization and
More information12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationSolving large Semidefinite Programs - Part 1 and 2
Solving large Semidefinite Programs - Part 1 and 2 Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria F. Rendl, Singapore workshop 2006 p.1/34 Overview Limits of Interior
More informationA new primal-dual path-following method for convex quadratic programming
Volume 5, N., pp. 97 0, 006 Copyright 006 SBMAC ISSN 00-805 www.scielo.br/cam A new primal-dual path-following method for convex quadratic programming MOHAMED ACHACHE Département de Mathématiques, Faculté
More informationCanonical Problem Forms. Ryan Tibshirani Convex Optimization
Canonical Problem Forms Ryan Tibshirani Convex Optimization 10-725 Last time: optimization basics Optimization terology (e.g., criterion, constraints, feasible points, solutions) Properties and first-order
More informationA NEW SECOND-ORDER CONE PROGRAMMING RELAXATION FOR MAX-CUT PROBLEMS
Journal of the Operations Research Society of Japan 2003, Vol. 46, No. 2, 164-177 2003 The Operations Research Society of Japan A NEW SECOND-ORDER CONE PROGRAMMING RELAXATION FOR MAX-CUT PROBLEMS Masakazu
More informationThe Simplest Semidefinite Programs are Trivial
The Simplest Semidefinite Programs are Trivial Robert J. Vanderbei Bing Yang Program in Statistics & Operations Research Princeton University Princeton, NJ 08544 January 10, 1994 Technical Report SOR-93-12
More informationThe LP approach potentially allows one to approximately solve large scale semidefinite programs using state of the art linear solvers. Moreover one ca
Semi-infinite linear programming approaches to semidefinite programming problems Λy Kartik Krishnan z John E. Mitchell x December 9, 2002 Abstract Interior point methods, the traditional methods for the
More informationE5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming
E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program
More informationAcyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs
2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca and Eilyan Bitar
More informationInterior Point Methods: Second-Order Cone Programming and Semidefinite Programming
School of Mathematics T H E U N I V E R S I T Y O H F E D I N B U R G Interior Point Methods: Second-Order Cone Programming and Semidefinite Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio
More informationAgenda. Interior Point Methods. 1 Barrier functions. 2 Analytic center. 3 Central path. 4 Barrier method. 5 Primal-dual path following algorithms
Agenda Interior Point Methods 1 Barrier functions 2 Analytic center 3 Central path 4 Barrier method 5 Primal-dual path following algorithms 6 Nesterov Todd scaling 7 Complexity analysis Interior point
More informationA direct formulation for sparse PCA using semidefinite programming
A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley A. d Aspremont, INFORMS, Denver,
More informationNew stopping criteria for detecting infeasibility in conic optimization
Optimization Letters manuscript No. (will be inserted by the editor) New stopping criteria for detecting infeasibility in conic optimization Imre Pólik Tamás Terlaky Received: March 21, 2008/ Accepted:
More informationRelaxations and Randomized Methods for Nonconvex QCQPs
Relaxations and Randomized Methods for Nonconvex QCQPs Alexandre d Aspremont, Stephen Boyd EE392o, Stanford University Autumn, 2003 Introduction While some special classes of nonconvex problems can be
More informationDegeneracy in Maximal Clique Decomposition for Semidefinite Programs
MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Degeneracy in Maximal Clique Decomposition for Semidefinite Programs Raghunathan, A.U.; Knyazev, A. TR2016-040 July 2016 Abstract Exploiting
More informationCroatian Operational Research Review (CRORR), Vol. 3, 2012
126 127 128 129 130 131 132 133 REFERENCES [BM03] S. Burer and R.D.C. Monteiro (2003), A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Mathematical Programming
More informationPENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP p.1/39
PENNON A Generalized Augmented Lagrangian Method for Convex NLP and SDP Michal Kočvara Institute of Information Theory and Automation Academy of Sciences of the Czech Republic and Czech Technical University
More informationHandout 6: Some Applications of Conic Linear Programming
ENGG 550: Foundations of Optimization 08 9 First Term Handout 6: Some Applications of Conic Linear Programming Instructor: Anthony Man Cho So November, 08 Introduction Conic linear programming CLP, and
More informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with M. Bomze (Wien) and F. Jarre (Düsseldorf) and
More informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with I.M. Bomze (Wien) and F. Jarre (Düsseldorf) IMA
More informationEE 227A: Convex Optimization and Applications October 14, 2008
EE 227A: Convex Optimization and Applications October 14, 2008 Lecture 13: SDP Duality Lecturer: Laurent El Ghaoui Reading assignment: Chapter 5 of BV. 13.1 Direct approach 13.1.1 Primal problem Consider
More informatione-companion ONLY AVAILABLE IN ELECTRONIC FORM
OPERATIONS RESEARCH doi 10.1287/opre.1110.0918ec e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 2011 INFORMS Electronic Companion Mixed Zero-One Linear Programs Under Objective Uncertainty: A Completely
More informationThere are several approaches to solve UBQP, we will now briefly discuss some of them:
3 Related Work There are several approaches to solve UBQP, we will now briefly discuss some of them: Since some of them are actually algorithms for the Max Cut problem (MC), it is necessary to show their
More informationSemidefinite Programming Basics and Applications
Semidefinite Programming Basics and Applications Ray Pörn, principal lecturer Åbo Akademi University Novia University of Applied Sciences Content What is semidefinite programming (SDP)? How to represent
More informationPrimal-Dual Interior-Point Methods. Javier Peña Convex Optimization /36-725
Primal-Dual Interior-Point Methods Javier Peña Convex Optimization 10-725/36-725 Last time: duality revisited Consider the problem min x subject to f(x) Ax = b h(x) 0 Lagrangian L(x, u, v) = f(x) + u T
More informationSDPARA : SemiDefinite Programming Algorithm PARAllel version
Research Reports on Mathematical and Computing Sciences Series B : Operations Research Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1 Oh-Okayama, Meguro-ku, Tokyo
More informationLMI MODELLING 4. CONVEX LMI MODELLING. Didier HENRION. LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ. Universidad de Valladolid, SP March 2009
LMI MODELLING 4. CONVEX LMI MODELLING Didier HENRION LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ Universidad de Valladolid, SP March 2009 Minors A minor of a matrix F is the determinant of a submatrix
More informationSparse PCA with applications in finance
Sparse PCA with applications in finance A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley Available online at www.princeton.edu/~aspremon 1 Introduction
More informationAgenda. Applications of semidefinite programming. 1 Control and system theory. 2 Combinatorial and nonconvex optimization
Agenda Applications of semidefinite programming 1 Control and system theory 2 Combinatorial and nonconvex optimization 3 Spectral estimation & super-resolution Control and system theory SDP in wide use
More informationA priori bounds on the condition numbers in interior-point methods
A priori bounds on the condition numbers in interior-point methods Florian Jarre, Mathematisches Institut, Heinrich-Heine Universität Düsseldorf, Germany. Abstract Interior-point methods are known to be
More informationThe Q Method for Symmetric Cone Programmin
The Q Method for Symmetric Cone Programming The Q Method for Symmetric Cone Programmin Farid Alizadeh and Yu Xia alizadeh@rutcor.rutgers.edu, xiay@optlab.mcma Large Scale Nonlinear and Semidefinite Progra
More informationLecture 14: Optimality Conditions for Conic Problems
EE 227A: Conve Optimization and Applications March 6, 2012 Lecture 14: Optimality Conditions for Conic Problems Lecturer: Laurent El Ghaoui Reading assignment: 5.5 of BV. 14.1 Optimality for Conic Problems
More informationSemidefinite Programming
Semidefinite Programming Basics and SOS Fernando Mário de Oliveira Filho Campos do Jordão, 2 November 23 Available at: www.ime.usp.br/~fmario under talks Conic programming V is a real vector space h, i
More informationAnalytic Center Cutting-Plane Method
Analytic Center Cutting-Plane Method S. Boyd, L. Vandenberghe, and J. Skaf April 14, 2011 Contents 1 Analytic center cutting-plane method 2 2 Computing the analytic center 3 3 Pruning constraints 5 4 Lower
More informationA direct formulation for sparse PCA using semidefinite programming
A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley Available online at www.princeton.edu/~aspremon
More informationThe Ongoing Development of CSDP
The Ongoing Development of CSDP Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.edu Joseph Young Department of Mathematics New Mexico Tech (Now at Rice University)
More informationIntroduction to Semidefinite Programming I: Basic properties a
Introduction to Semidefinite Programming I: Basic properties and variations on the Goemans-Williamson approximation algorithm for max-cut MFO seminar on Semidefinite Programming May 30, 2010 Semidefinite
More informationc 2000 Society for Industrial and Applied Mathematics
SIAM J. OPIM. Vol. 10, No. 3, pp. 750 778 c 2000 Society for Industrial and Applied Mathematics CONES OF MARICES AND SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES MASAKAZU KOJIMA AND LEVEN UNÇEL Abstract.
More information1 Introduction Semidenite programming (SDP) has been an active research area following the seminal work of Nesterov and Nemirovski [9] see also Alizad
Quadratic Maximization and Semidenite Relaxation Shuzhong Zhang Econometric Institute Erasmus University P.O. Box 1738 3000 DR Rotterdam The Netherlands email: zhang@few.eur.nl fax: +31-10-408916 August,
More informationא K ٢٠٠٢ א א א א א א E٤
المراجع References المراجع العربية K ١٩٩٠ א א א א א E١ K ١٩٩٨ א א א E٢ א א א א א E٣ א K ٢٠٠٢ א א א א א א E٤ K ٢٠٠١ K ١٩٨٠ א א א א א E٥ المراجع الا جنبية [AW] [AF] [Alh] [Ali1] [Ali2] S. Al-Homidan and
More informationIE 521 Convex Optimization
Lecture 14: and Applications 11th March 2019 Outline LP SOCP SDP LP SOCP SDP 1 / 21 Conic LP SOCP SDP Primal Conic Program: min c T x s.t. Ax K b (CP) : b T y s.t. A T y = c (CD) y K 0 Theorem. (Strong
More informationSecond Order Cone Programming Relaxation of Nonconvex Quadratic Optimization Problems
Second Order Cone Programming Relaxation of Nonconvex Quadratic Optimization Problems Sunyoung Kim Department of Mathematics, Ewha Women s University 11-1 Dahyun-dong, Sudaemoon-gu, Seoul 120-750 Korea
More informationCSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming
More informationCSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization
CSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization April 6, 2018 1 / 34 This material is covered in the textbook, Chapters 9 and 10. Some of the materials are taken from it. Some of
More informationIntroduction to Semidefinite Programs
Introduction to Semidefinite Programs Masakazu Kojima, Tokyo Institute of Technology Semidefinite Programming and Its Application January, 2006 Institute for Mathematical Sciences National University of
More informationResearch overview. Seminar September 4, Lehigh University Department of Industrial & Systems Engineering. Research overview.
Research overview Lehigh University Department of Industrial & Systems Engineering COR@L Seminar September 4, 2008 1 Duality without regularity condition Duality in non-exact arithmetic 2 interior point
More informationSEMIDEFINITE PROGRAM BASICS. Contents
SEMIDEFINITE PROGRAM BASICS BRIAN AXELROD Abstract. A introduction to the basics of Semidefinite programs. Contents 1. Definitions and Preliminaries 1 1.1. Linear Algebra 1 1.2. Convex Analysis (on R n
More informationRelations between Semidefinite, Copositive, Semi-infinite and Integer Programming
Relations between Semidefinite, Copositive, Semi-infinite and Integer Programming Author: Faizan Ahmed Supervisor: Dr. Georg Still Master Thesis University of Twente the Netherlands May 2010 Relations
More informationHandout 8: Dealing with Data Uncertainty
MFE 5100: Optimization 2015 16 First Term Handout 8: Dealing with Data Uncertainty Instructor: Anthony Man Cho So December 1, 2015 1 Introduction Conic linear programming CLP, and in particular, semidefinite
More informationInfeasible Primal-Dual (Path-Following) Interior-Point Methods for Semidefinite Programming*
Infeasible Primal-Dual (Path-Following) Interior-Point Methods for Semidefinite Programming* Yin Zhang Dept of CAAM, Rice University Outline (1) Introduction (2) Formulation & a complexity theorem (3)
More informationUsing quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints
Noname manuscript No. (will be inserted by the editor) Using quadratic conve reformulation to tighten the conve relaation of a quadratic program with complementarity constraints Lijie Bai John E. Mitchell
More informationInfeasible Primal-Dual (Path-Following) Interior-Point Methods for Semidefinite Programming*
Infeasible Primal-Dual (Path-Following) Interior-Point Methods for Semidefinite Programming* Notes for CAAM 564, Spring 2012 Dept of CAAM, Rice University Outline (1) Introduction (2) Formulation & a complexity
More informationConvex Optimization and l 1 -minimization
Convex Optimization and l 1 -minimization Sangwoon Yun Computational Sciences Korea Institute for Advanced Study December 11, 2009 2009 NIMS Thematic Winter School Outline I. Convex Optimization II. l
More informationA smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function
Volume 30, N. 3, pp. 569 588, 2011 Copyright 2011 SBMAC ISSN 0101-8205 www.scielo.br/cam A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister
More informationSDP Relaxations for MAXCUT
SDP Relaxations for MAXCUT from Random Hyperplanes to Sum-of-Squares Certificates CATS @ UMD March 3, 2017 Ahmed Abdelkader MAXCUT SDP SOS March 3, 2017 1 / 27 Overview 1 MAXCUT, Hardness and UGC 2 LP
More informationA Hierarchy of Polyhedral Approximations of Robust Semidefinite Programs
A Hierarchy of Polyhedral Approximations of Robust Semidefinite Programs Raphael Louca Eilyan Bitar Abstract Robust semidefinite programs are NP-hard in general In contrast, robust linear programs admit
More informationA Continuation Approach Using NCP Function for Solving Max-Cut Problem
A Continuation Approach Using NCP Function for Solving Max-Cut Problem Xu Fengmin Xu Chengxian Ren Jiuquan Abstract A continuous approach using NCP function for approximating the solution of the max-cut
More informationPositive semidefinite rank
1/15 Positive semidefinite rank Hamza Fawzi (MIT, LIDS) Joint work with João Gouveia (Coimbra), Pablo Parrilo (MIT), Richard Robinson (Microsoft), James Saunderson (Monash), Rekha Thomas (UW) DIMACS Workshop
More informationInterior Point Methods. We ll discuss linear programming first, followed by three nonlinear problems. Algorithms for Linear Programming Problems
AMSC 607 / CMSC 764 Advanced Numerical Optimization Fall 2008 UNIT 3: Constrained Optimization PART 4: Introduction to Interior Point Methods Dianne P. O Leary c 2008 Interior Point Methods We ll discuss
More information12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationCS295: Convex Optimization. Xiaohui Xie Department of Computer Science University of California, Irvine
CS295: Convex Optimization Xiaohui Xie Department of Computer Science University of California, Irvine Course information Prerequisites: multivariate calculus and linear algebra Textbook: Convex Optimization
More informationDimension reduction for semidefinite programming
1 / 22 Dimension reduction for semidefinite programming Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology
More informationIII. Applications in convex optimization
III. Applications in convex optimization nonsymmetric interior-point methods partial separability and decomposition partial separability first order methods interior-point methods Conic linear optimization
More informationFinding a Strict Feasible Dual Initial Solution for Quadratic Semidefinite Programming
Applied Mathematical Sciences, Vol 6, 22, no 4, 229-234 Finding a Strict Feasible Dual Initial Solution for Quadratic Semidefinite Programming El Amir Djeffal Departement of Mathematics University of Batna,
More informationLecture 3: Semidefinite Programming
Lecture 3: Semidefinite Programming Lecture Outline Part I: Semidefinite programming, examples, canonical form, and duality Part II: Strong Duality Failure Examples Part III: Conditions for strong duality
More informationCourse Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)
Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization:
More informationLecture: Introduction to LP, SDP and SOCP
Lecture: Introduction to LP, SDP and SOCP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2015.html wenzw@pku.edu.cn Acknowledgement:
More informationIdentifying Redundant Linear Constraints in Systems of Linear Matrix. Inequality Constraints. Shafiu Jibrin
Identifying Redundant Linear Constraints in Systems of Linear Matrix Inequality Constraints Shafiu Jibrin (shafiu.jibrin@nau.edu) Department of Mathematics and Statistics Northern Arizona University, Flagstaff
More information15. Conic optimization
L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 15-1 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone
More informationA solution approach for linear optimization with completely positive matrices
A solution approach for linear optimization with completely positive matrices Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with M. Bomze (Wien) and F.
More informationOutline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lagrangian Relaxation Lecture 12 Single Machine Models, Column Generation 2. Dantzig-Wolfe Decomposition Dantzig-Wolfe Decomposition Delayed Column
More informationLecture 7: Convex Optimizations
Lecture 7: Convex Optimizations Radu Balan, David Levermore March 29, 2018 Convex Sets. Convex Functions A set S R n is called a convex set if for any points x, y S the line segment [x, y] := {tx + (1
More informationLecture 9: Dantzig-Wolfe Decomposition
Lecture 9: Dantzig-Wolfe Decomposition (3 units) Outline Dantzig-Wolfe decomposition Column generation algorithm Relation to Lagrangian dual Branch-and-price method Generated assignment problem and multi-commodity
More informationLecture: Algorithms for LP, SOCP and SDP
1/53 Lecture: Algorithms for LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html wenzw@pku.edu.cn Acknowledgement:
More informationCuts for Conic Mixed-Integer Programming
Cuts for Conic Mixed-Integer Programming Alper Atamtürk and Vishnu Narayanan Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720-1777 USA atamturk@berkeley.edu,
More informationSPARSE SECOND ORDER CONE PROGRAMMING FORMULATIONS FOR CONVEX OPTIMIZATION PROBLEMS
Journal of the Operations Research Society of Japan 2008, Vol. 51, No. 3, 241-264 SPARSE SECOND ORDER CONE PROGRAMMING FORMULATIONS FOR CONVEX OPTIMIZATION PROBLEMS Kazuhiro Kobayashi Sunyoung Kim Masakazu
More informationA PREDICTOR-CORRECTOR PATH-FOLLOWING ALGORITHM FOR SYMMETRIC OPTIMIZATION BASED ON DARVAY'S TECHNIQUE
Yugoslav Journal of Operations Research 24 (2014) Number 1, 35-51 DOI: 10.2298/YJOR120904016K A PREDICTOR-CORRECTOR PATH-FOLLOWING ALGORITHM FOR SYMMETRIC OPTIMIZATION BASED ON DARVAY'S TECHNIQUE BEHROUZ
More information