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An unconstrained smooth minimization reformulation of the secondorder cone complementarity problem
 MATH. PROGRAM., SER. B 104, 293–327 (2005)
, 2005
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Analysis of Nonsmooth VectorValued Functions Associated with SecondOrder Cones
, 2002
"... Let K be the Lorentz/secondorder cone in IR . For any function f from IR to IR, one can de ne a corresponding function f (x) on IR by applying f to the spectral values of the spectral decomposition of x 2 IR . We show that this vectorvalued function inherits from f the properties of continuity, (l ..."
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Cited by 30 (7 self)
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Let K be the Lorentz/secondorder cone in IR . For any function f from IR to IR, one can de ne a corresponding function f (x) on IR by applying f to the spectral values of the spectral decomposition of x 2 IR . We show that this vectorvalued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as (rhoorder) semismoothness. These results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving secondorder cone programs and complementarity problems.
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 29 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.
A masked spectral bound for maximumentropy sampling
 In: mODa 7–Advances in ModelOriented Design and Analysis, Contributions to Statistics
, 2004
"... LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its ..."
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Cited by 8 (1 self)
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LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). Copies may be requested from IBM T. J. Watson Research
Interior point and semidefinite approaches in combinatorial optimization
, 2005
"... Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient ..."
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Cited by 8 (4 self)
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Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primaldual interiorpoint methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NPhard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to nonconvex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include nonconvex potential reduction methods, interior point cutting plane methods, primaldual IPMs and firstorder algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.
On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász’s Theta Function
, 2004
"... We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász’s theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its subgraphs. We develop a ..."
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Cited by 5 (1 self)
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We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász’s theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its subgraphs. We develop an efficient, polynomialtime algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warmstart strategy
The StateoftheArt in Conic Optimization Software
"... Abstract This work gives an overview of the major codes available for the solution of linear semidefinite (SDP) and secondorder cone (SOCP) programs. Many of these codes also solve linear programs (LP). Some developments since the 7th DIMACS Challenge [10, 18] are pointed out as well as some curren ..."
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Abstract This work gives an overview of the major codes available for the solution of linear semidefinite (SDP) and secondorder cone (SOCP) programs. Many of these codes also solve linear programs (LP). Some developments since the 7th DIMACS Challenge [10, 18] are pointed out as well as some currently under way. Instead of presenting performance tables, reference is made to the ongoing benchmark webpage [20] as well as other related efforts. 1
Alternative proofs for some results of vectorvalued functions associated with secondorder cones
 Journal of Nonlinear and Convex Analysis
"... Abstract. Let Kn be the Lorentz/secondorder cone in IRn. For any function f from IR to IR, one can define a corresponding vectorvalued function f soc (x) on IRn by applying f to the spectral values of the spectral decomposition of x ∈ IRn with respect to Kn. It was shown by J.S. Chen, X. Chen and ..."
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Abstract. Let Kn be the Lorentz/secondorder cone in IRn. For any function f from IR to IR, one can define a corresponding vectorvalued function f soc (x) on IRn by applying f to the spectral values of the spectral decomposition of x ∈ IRn with respect to Kn. It was shown by J.S. Chen, X. Chen and P. Tseng that this vectorvalued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. It was also proved by D. Sun and J. Sun that the vectorvalued FischerBurmeister function associated with secondorder cone is strongly semismooth. All proofs for the above results are based on a special relation between the vectorvalued function and the matrixvalued function over symmetric matrices. In this paper, we provide a straightforward and intuitive way to prove all the above results by using the simple structure of secondorder cone and spectral decomposition. Key words. Secondorder cone, vectorvalued function, semismooth function, comple
Using interiorpoint methods within an outer approximation framework for mixedinteger nonlinear programming
 IMAMINLP Issue
"... Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via o ..."
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Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via outer approximation. However, traditionally, infeasible primaldual interiorpoint methods have had two main perceived deficiencies: (1) lack of infeasibility detection capabilities, and (2) poor performance after a warmstart. In this paper, we propose the exact primaldual penalty approach as a means to overcome these deficiencies. The generality of this approach to handle any change to the problem makes it suitable for the outer approximation framework, where each nonlinear subproblem can differ from the others in the sequence in a variety of ways. Additionally, we examine cases where the nonlinear subproblems take on special forms, namely those of secondorder cone programming problems and semidefinite programming problems. Encouraging numerical results are provided.