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DETERMINANT MAXIMIZATION WITH LINEAR MATRIX INEQUALITY CONSTRAINTS
"... The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the s ..."
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Cited by 200 (18 self)
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The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the semidefinite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interiorpoint method, with a simplified analysis of the worstcase complexity and numerical results that indicate that the method is very efficient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interiorpoint method will generally be slower; the advantage is that it handles a much wider variety of problems.
How good are interior point methods? KleeMinty cubes tighten iterationcomplexity bounds
, 2004
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A Generic PathFollowing Algorithm With a Sliding Constraint and Its Application to Linear Programming and the Computation of Analytic Centers
, 1996
"... We propose a generic pathfollowing scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic pr ..."
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Cited by 8 (5 self)
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We propose a generic pathfollowing scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic procedure in use, primal, dual or primaldual. We show convergence in O( p n) iterations. We verify that the primal, dual and primaldual algorithms satisfy the local quadratic convergence property. The method can be applied to solve the linear programming problem (with a feasible start) and to compute the analytic center of a bounded polytope. The generic pathfollowing scheme easily extends to the logarithmic penalty barrier approach. Keywords Interior point method, method of centers, pathfollowing, linear programming. This work has been completed with the support from the Swiss National Foundation for Scientific Research, grant 1234002.92. 1 Introduction Shortly after Karmarkar's s...
Polynomial Cutting Plane Algorithms for TwoStage Stochastic Linear Programs Based on Ellipsoids, Volumetric Centers and Analytic Centers
 WASHINGTON STATE UNIVERSITY
, 1996
"... Traditional simplexbased algorithms for twostage stochastic linear programscan be broadly divided into two classes: (a) those that explicitly exploit the structure of the equivalent largescale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly ..."
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Traditional simplexbased algorithms for twostage stochastic linear programscan be broadly divided into two classes: (a) those that explicitly exploit the structure of the equivalent largescale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly exploit that structure. Algorithms of class (b) are in general preferred. In 1988, following the work of Karmarkar for general linear programs, Birge and Qi [10] proposed a specialization of Karmarkar's algorithm for twostage stochastic linear programs. The algorithm of Birge and Qi [10] is the first interior point analog of class (a). Several other authors have studied related and different interior point analogs of class (a). Birge and Qi [10] also presented an analysis of the computational complexity of their algorithm. This analysis indicates that the computational complexity (in terms of total arithmetic operations) of their algorithm is in general smaller than that of the Karmarkar's ...
TimeRecursive VelocityAdapted SpatioTemporal ScaleSpace Filters
 In Proc. ECCV, volume 2350 of LNCS
, 2002
"... This paper presents a theory for constructing and computing velocityadapted scalespace filters for spariotemporal image data. ..."
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Cited by 7 (6 self)
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This paper presents a theory for constructing and computing velocityadapted scalespace filters for spariotemporal image data.
Finding a point in the relative interior of a polyhedron
, 2007
"... A new initialization or ‘Phase I ’ strategy for feasible interior point methods for linear programming is proposed that computes a point on the primaldual central path associated with the linear program. Provided there exist primaldual strictly feasible points — an allpervasive assumption in inte ..."
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Cited by 6 (3 self)
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A new initialization or ‘Phase I ’ strategy for feasible interior point methods for linear programming is proposed that computes a point on the primaldual central path associated with the linear program. Provided there exist primaldual strictly feasible points — an allpervasive assumption in interior point method theory that implies the existence of the central path — our initial method (Algorithm 1) is globally Qlinearly and asymptotically Qquadratically convergent, with a provable worstcase iteration complexity bound. When this assumption is not met, the numerical behaviour of Algorithm 1 is highly disappointing, even when the problem is primaldual feasible. This is due to the presence of implicit equalities, inequality constraints that hold as equalities at all the feasible points. Controlled perturbations of the inequality constraints of the primaldual problems are introduced — geometrically equivalent to enlarging the primaldual feasible region and then systematically contracting it back to its initial shape — in order for the perturbed problems to satisfy the assumption. Thus Algorithm 1 can successfully be employed to solve each of the perturbed problems. We show that, when there exist primaldual strictly feasible points of the original problems, the resulting method, Algorithm 2, finds such a point in a finite number of changes to the perturbation parameters. When implicit equalities are present, but the original problem and its dual are feasible, Algorithm 2 asymptotically detects all the primaldual implicit equalities and generates a point in the relative interior of the primaldual feasible set. Algorithm 2 can also asymptotically detect primaldual infeasibility. Successful numerical experience with Algorithm 2 on linear programs from NETLIB and CUTEr, both with and without any significant preprocessing of the problems, indicates that Algorithm 2 may be used as an algorithmic preprocessor for removing implicit equalities, with theoretical guarantees of convergence. 1
The longstep method of analytic centers for fractional problems
 Mathematical Programming
, 1997
"... We develop a longstep surfacefollowing version of the method of analytic centers for the fractionallinear problem min {t0  t0B(x) − A(x) ∈ H, B(x) ∈ K, x ∈ G}, where H is a closed convex domain, K is a convex cone contained in the recessive cone of H, G is a convex domain and B(·), A(·) are a ..."
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We develop a longstep surfacefollowing version of the method of analytic centers for the fractionallinear problem min {t0  t0B(x) − A(x) ∈ H, B(x) ∈ K, x ∈ G}, where H is a closed convex domain, K is a convex cone contained in the recessive cone of H, G is a convex domain and B(·), A(·) are affine mappings. Tracing a twodimensional surface of analytic centers rather than the usual path of centers allows to skip the initial “centering ” phase of the pathfollowing scheme. The proposed longstep policy of tracing the surface fits the best known overall polynomialtime complexity bounds for the method and, at the same time, seems to be more attractive computationally than the shortstep policy, which was previously the only one giving good complexity bounds. 1