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Complexity of Convex Optimization Using GeometryBased Measures and a Reference Point
, 2002
"... Our concern lies in solving the following convex optimization problem: G P : minimize x c where P is a closed convex subset of the ndimensional vector space X. We bound the complexity of computing an almostoptimal solution of G P in terms of natural geometrybased measures of the feasible ..."
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Cited by 17 (7 self)
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Our concern lies in solving the following convex optimization problem: G P : minimize x c where P is a closed convex subset of the ndimensional vector space X. We bound the complexity of computing an almostoptimal solution of G P in terms of natural geometrybased measures of the feasible region and the levelset of almostoptimal solutions, relative to a given that might be close to the feasible region and/or the almostoptimal level set. This contrasts with other complexity bounds for convex optimization that rely on databased condition numbers or algebraic measures, and that do not take into account any a priori reference point information. AMS Subject Classification: 90C, 90C05, 90C60 Keywords: Convex Optimization, Complexity, InteriorPoint Method, Barrier Method This research has been partially supported through the SingaporeMIT Alliance. Portions of this research were undertaken when the author was a Visiting Scientist at Delft University of Technology.
Behavioral measures and their correlation with IPM iteration counts on semidefinite programming problems
, 2005
"... We study four measures of problem instance behavior that might account for the observed differences in interiorpoint method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasib ..."
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Cited by 9 (1 self)
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We study four measures of problem instance behavior that might account for the observed differences in interiorpoint method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar) condition measure C(d) of the data instance, (iii) a measure of the nearabsence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the correlation between these measures and IPM iteration counts (solved using the software SDPT3) when the measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.896), and is a very good predictor of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The nearabsence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.
On the behavior of the homogeneous selfdual model for conic convex optimization
 MIT Operations Research
, 2004
"... There is a natural norm associated with a starting point of the homogeneous selfdual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of εoptimal solutions, and (ii ..."
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Cited by 3 (1 self)
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There is a natural norm associated with a starting point of the homogeneous selfdual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of εoptimal solutions, and (ii) the maximum distance of εoptimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stoppingrule theory for HSDbased interiorpoint methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the εoptimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous selfdual model that might improve the resulting solution time in practice.
PROJECTIVE PRECONDITIONERS FOR IMPROVING THE BEHAVIOR OF A HOMOGENEOUS CONIC LINEAR System
"... In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax = 0, ¯s T x = 1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v: ¯s − AT v ∈ C ∗}. Such a projective transformation serves to ..."
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Cited by 3 (1 self)
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In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax = 0, ¯s T x = 1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v: ¯s − AT v ∈ C ∗}. Such a projective transformation serves to precondition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry of ˆv in H ◦ ¯s as well as the complexity parameter ϑ of the barrier for C. Under the assumption that F has an interior solution, H ◦ ¯s must contain a point v whose symmetry is at least 1/m; if we can find a point whose symmetry is Ω(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be stronglypolynomialtime in m and ϑ. We present a method for generating such a point ˆv based on sampling and on a geometric random walk on H ◦ ¯s with associated complexity and probabilistic analysis. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective preconditioning methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46 % decrease in average IPM iterations for 100 randomly generated poorlybehaved problem instances of dimension 1000 × 5000.
Condition and Complexity Measures for Infeasibility Certificates of Systems of Linear Inequalities and Their Sensitivity Analysis
, 2002
"... We begin with a study of the infeasibility measures for linear programming problems. For this purpose, we consider feasibility problems in Karmarkar's standard form. Our main focus is on the complexity measures which can be used to bound the amount of computational effort required to solve syst ..."
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Cited by 1 (1 self)
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We begin with a study of the infeasibility measures for linear programming problems. For this purpose, we consider feasibility problems in Karmarkar's standard form. Our main focus is on the complexity measures which can be used to bound the amount of computational effort required to solve systems of linear inequalities and related problems in certain ways.
Epigraphical cones I
 JOURNAL OF CONVEX ANALYSIS, 2011, IN PRESS
"... Up to orthogonal transformation, a solid closed convex cone K in the Euclidean space Rn+1 is the epigraph of a nonnegative sublinear function f: Rn → R. This work explores the link between the geometric properties of K and the analytic properties of f. ..."
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Cited by 1 (1 self)
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Up to orthogonal transformation, a solid closed convex cone K in the Euclidean space Rn+1 is the epigraph of a nonnegative sublinear function f: Rn → R. This work explores the link between the geometric properties of K and the analytic properties of f.
Key words. Convex Optimization – Convex Cone – Conic Optimization – Duality – Level Sets – SelfDual Embedding – Selfscaled Cone 1. Preliminaries
, 2005
"... Abstract. There is a natural norm associated with a starting point of the homogeneous selfdual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of εoptimal solutions ..."
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Abstract. There is a natural norm associated with a starting point of the homogeneous selfdual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of εoptimal solutions, and (ii) the maximum distance of εoptimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stoppingrule theory for HSDbased interiorpoint solvers such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the εoptimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous selfdual model that might improve the resulting solution time in practice.
Correlation with IPM Iteration Counts on SemiDefinite Programming Problems
"... This paper is dedicated in memory of Jos Sturm ..."
PROJECTIVE PRECONDITIONERS FOR IMPROVING THE BEHAVIOR OF A HOMOGENEOUS CONIC LINEAR
, 2005
"... Abstract. In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax =0, ¯s T x =1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v:¯s − AT v ∈ C ∗}. Such a projective transformation se ..."
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Abstract. In this paper we present a general theory for transforming a normalized homogeneous conic system F: Ax =0, ¯s T x =1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point ˆv in the set H ◦ ¯s = {v:¯s − AT v ∈ C ∗}. Such a projective transformation serves to precondition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry of ˆv in H ◦ ¯s as well as the complexity parameter ϑ of the barrier for C. Under the assumption that F has an interior solution, H ◦ ¯s must contain a point v whose symmetry is at least 1/m; if we can find a point whose symmetry is Ω(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be stronglypolynomialtime in m and ϑ. We present a method for generating such a point ˆv based on sampling and on a geometric random walk on H ◦ ¯s with associated complexity and probabilistic analysis. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective preconditioning methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46 % decrease in average IPM iterations for 100 randomly generated poorlybehaved problem instances of dimension 1000 × 5000. 1.