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SecondOrder Cone Programming
 MATHEMATICAL PROGRAMMING
, 2001
"... In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic struc ..."
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Cited by 247 (11 self)
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In this paper we survey the second order cone programming problem (SOCP). First we present several applications of the problem in various areas of engineering and robust optimization problems. We also give examples of optimization problems that can be cast as SOCPs. Next we review an algebraic structure that is connected to SOCP. This algebra is a special case of a Euclidean Jordan algebra. After presenting duality theory, complementary slackness conditions, and definitions and algebraic characterizations of primal and dual nondegeneracy and strict complementarity we review the logarithmic barrier function for the SOCP problem and survey the pathfollowing interior point algorithms for it. Next we examine numerically stable methods for solving the interior point methods and study ways that sparsity in the input data can be exploited. Finally we give some current and future research direction in SOCP.
Spectral bounds for sparse PCA: Exact and greedy algorithms
 Advances in Neural Information Processing Systems 18
, 2006
"... Sparse PCA seeks approximate sparse “eigenvectors ” whose projections capture the maximal variance of data. As a cardinalityconstrained and nonconvex optimization problem, it is NPhard and yet it is encountered in a wide range of applied fields, from bioinformatics to finance. Recent progress ha ..."
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Cited by 79 (4 self)
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Sparse PCA seeks approximate sparse “eigenvectors ” whose projections capture the maximal variance of data. As a cardinalityconstrained and nonconvex optimization problem, it is NPhard and yet it is encountered in a wide range of applied fields, from bioinformatics to finance. Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint. In contrast, we consider an alternative discrete spectral formulation based on variational eigenvalue bounds and provide an effective greedy strategy as well as provably optimal solutions using branchandbound search. Moreover, the exact methodology used reveals a simple renormalization step that improves approximate solutions obtained by any continuous method. The resulting performance gain of discrete algorithms is demonstrated on realworld benchmark data and in extensive Monte Carlo evaluation trials. 1
Testing the Nullspace Property using Semidefinite Programming
, 2009
"... Recent results in compressed sensing show that, under certain conditions, the sparsest solution to an underdetermined set of linear equations can be recovered by solving a linear program. These results either rely on computing sparse eigenvalues of the design matrix or on properties of its nullspace ..."
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Cited by 37 (1 self)
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Recent results in compressed sensing show that, under certain conditions, the sparsest solution to an underdetermined set of linear equations can be recovered by solving a linear program. These results either rely on computing sparse eigenvalues of the design matrix or on properties of its nullspace. So far, no tractable algorithm is known to test these conditions and most current results rely on asymptotic properties of random matrices. Given a matrix A, we use semidefinite relaxation techniques to test the nullspace property on A and show on some numerical examples that these relaxation bounds can prove perfect recovery of sparse solutions with relatively high cardinality.
Subspace identification with guaranteed stability using constrained optimization
 IEEE Transactions on Automatic Control
, 2003
"... Abstract—In system identification, the true system is often known to be stable. However, due to finite sample constraints, modeling errors, plant disturbances and measurement noise, the identified model may be unstable. We present a constrained optimization method to ensure asymptotic stability of ..."
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Cited by 22 (2 self)
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Abstract—In system identification, the true system is often known to be stable. However, due to finite sample constraints, modeling errors, plant disturbances and measurement noise, the identified model may be unstable. We present a constrained optimization method to ensure asymptotic stability of the identified model in the context of subspace identification methods. In subspace identification, we first obtain an estimate of the state sequence or extended observability matrix and then solve a least squares optimization problem to estimate the system parameters. To ensure asymptotic stability of the identified model, we write the leastsquares optimization problem as a convex linear programming problem with mixed equality, quadratic, and positive–semidefinite constraints suitable for existing convex optimization codes such as SeDuMi. We present examples to illustrate the method and compare to existing approaches. Index Terms—System identification, stability, subspace identification, convex optimization, linear systems. I.
Solution methodologies for the smallest enclosing circle problem
 Computational Optimization and Applications
"... Tribute. We would like to dedicate this paper to Elijah Polak. Professor Polak has made substantial contributions to a truly broad spectrum of topics in nonlinear optimization, including optimization for engineering design centering, multicriteria optimization, optimal control, feasible directions ..."
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Cited by 15 (0 self)
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Tribute. We would like to dedicate this paper to Elijah Polak. Professor Polak has made substantial contributions to a truly broad spectrum of topics in nonlinear optimization, including optimization for engineering design centering, multicriteria optimization, optimal control, feasible directions methods, quasiNewton and Newton methods, nondifferential optimization, semiinfinite optimization, conjugate directions methods, gradient projection and reduced gradient methods, and barrier methods, among many other topics. His many and varied contributions to our field are important today and will influence the research in our field well into the future. Abstract. Given a set of circles C = {c1,..., cn} on the Euclidean plane with centers {(a1,b1),..., (an,bn)} and radii {r1,..., rn}, the smallest enclosing circle (of fixed circles) problem is to find the circle of minimum radius that encloses all circles in C. We survey four known approaches for this problem, including a second order cone reformulation, a subgradient approach, a quadratic programming scheme, and a randomized incremental algorithm. For the last algorithm we also give some implementation details. It turns out the quadratic programming scheme outperforms the other three in our computational experiment.
A New Secondorder Cone Programming Relaxation for MAXCUT Problems
 Journal of Operations Research of Japan
, 2001
"... We propose a new relaxation scheme for the MAXCUT problem using secondorder cone programming. We construct relaxation problems to reflect the structure of the original graph. Numerical experiments show that our relaxation gives better bounds than those based on the spectral decomposition proposed ..."
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Cited by 10 (1 self)
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We propose a new relaxation scheme for the MAXCUT problem using secondorder cone programming. We construct relaxation problems to reflect the structure of the original graph. Numerical experiments show that our relaxation gives better bounds than those based on the spectral decomposition proposed by Kim and Kojima [16], and that the efficiency of the branchandbound method using our relaxation is comparable to that using semidefinite relaxation in some cases.
Interest Rate Model Calibration Using Semidefinite Programming
, 2003
"... We show that, for the purpose of pricing Swaptions, the Swap rate and the corresponding Forward rates can be considered lognormal under a single martingale measure. Swaptions can then be priced as options on a basket of lognormal assets and an approximation formula is derived for such options. This ..."
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Cited by 6 (0 self)
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We show that, for the purpose of pricing Swaptions, the Swap rate and the corresponding Forward rates can be considered lognormal under a single martingale measure. Swaptions can then be priced as options on a basket of lognormal assets and an approximation formula is derived for such options. This formula is centered around a BlackScholes price with an appropriate volatility, plus a correction term that can be interpreted as the expected tracking error. The calibration problem can then be solved very efficiently using semidefinite programming.
RiskManagement Methods for the Libor Market Model Using Semidefinite Programming
, 2003
"... When interest rate dynamics are described by the Libor Market Model as in Brace, Gatarek & Musiela (1997), we show how some essential riskmanagement results can be obtained from the dual of the calibration program. In particular, if the objetive is to maximize another swaption's price, we ..."
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Cited by 6 (0 self)
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When interest rate dynamics are described by the Libor Market Model as in Brace, Gatarek & Musiela (1997), we show how some essential riskmanagement results can be obtained from the dual of the calibration program. In particular, if the objetive is to maximize another swaption's price, we show that the optimal dual variables describe a hedging portfolio in the sense of Avellaneda & Paras (1996). In the general case, the local sensitivity of the covariance matrix to all market movement scenarios can be directly computed from the optimal dual solution. We also show how semidefinite programming can be used to manage the Gamma exposure of a portfolio.