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58
On the Expected Condition Number of Linear Programming Problems
 NUMER. MATH
, 2001
"... Let A be an n m real matrix and consider the linear conic system Ax 0; x 6= 0: In [Cheung and Cucker 1999] a condition number C (A) for this system is defined. In this paper we let the coefficients of A be independent identically distributed random variables with standard Gaussian distribution a ..."
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Cited by 24 (12 self)
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Let A be an n m real matrix and consider the linear conic system Ax 0; x 6= 0: In [Cheung and Cucker 1999] a condition number C (A) for this system is defined. In this paper we let the coefficients of A be independent identically distributed random variables with standard Gaussian distribution and we estimate the moments of the random variable ln C (A). In particular, when n is sufficiently larger than m we obtain for its expected value E(ln C (A)) = maxfln m; ln ln ng + O(1). Bounds for the expected value of the condition number introduced by Renegar [1994b, 1995a, 1995b] follow.
The Simplex and PolicyIteration Methods are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate
, 2010
"... We prove that the classic policyiteration method (Howard 1960), including the Simplex method (Dantzig 1947) with the mostnegativereducedcost pivoting rule, is a strongly polynomialtime algorithm for solving the Markov decision problem (MDP) with a fixed discount rate. Furthermore, the computati ..."
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Cited by 22 (1 self)
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We prove that the classic policyiteration method (Howard 1960), including the Simplex method (Dantzig 1947) with the mostnegativereducedcost pivoting rule, is a strongly polynomialtime algorithm for solving the Markov decision problem (MDP) with a fixed discount rate. Furthermore, the computational complexity of the policyiteration method (including the Simplex method) is superior to that of the only known strongly polynomialtime interiorpoint algorithm ([28] 2005) for solving this problem. The result is surprising since the Simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming (LP) problem, the Simplex (or simple policyiteration) method with the smallestindex pivoting rule was shown to be exponential for solving an MDP regardless of discount rates, and the policyiteration method was recently shown to be exponential for solving a undiscounted MDP. We also extend the result to solving MDPs with substochastic and transient state transition probability matrices. 1 Introduction of the Markov decision problem and
A New Condition Measure, PreConditioners, and Relations between Different Measures of Conditioning for Conic Linear Systems
, 2001
"... In recent years, a body of research into "condition numbers" for convex optimization has been developed, aimed at capturing the intuitive notion of problem behavior. This research has been shown to be relevant in studying the efficiency of algorithms (including interiorpoint algorithms) f ..."
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Cited by 22 (7 self)
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In recent years, a body of research into "condition numbers" for convex optimization has been developed, aimed at capturing the intuitive notion of problem behavior. This research has been shown to be relevant in studying the efficiency of algorithms (including interiorpoint algorithms) for convex optimization as well as other behavioral characteristics of these problems such as problem geometry, deformation under data perturbation, etc. This paper studies measures of conditioning for a conic linear system of the form (FP d ): Ax = b; x 2 CX , whose data is d = (A; b). We present a new measure of conditioning, denoted d , and we show implications of d for problem geometry and algorithm complexity, and demonstrate that the value of = d is independent of the speci c data representation of (FP d ). We then prove certain relations among a variety of condition measures for (FP d ), including d , d , d , and C(d). We discuss some drawbacks of using the condition number C(d) as the sole measure of conditioning of a conic linear system, and we introduce the notion of a "preconditioner" for (FP d ) which results in an equivalent formulation (FP ~ d ) of (FP d ) with a better condition number C( ~ d). We characterize the best such preconditioner and provide an algorithm and complexity analysis for constructing an equivalent data instance ~ d whose condition number C( ~ d) is within a known factor of the best possible.
Advances in convex optimization: Conic programming
 In Proceedings of International Congress of Mathematicians
, 2007
"... Abstract. During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit ..."
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Abstract. During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of the resulting theory (conic duality and primaldual interior point polynomial time algorithms), outline the extremely rich “expressive abilities ” of conic quadratic and semidefinite programming and discuss a number of instructive applications.
An iterative solverbased infeasible primaldual pathfollowing algorithm for convex quadratic programming
 SIAM J. OPTIM
, 2006
"... In this paper we develop a longstep primaldual infeasible pathfollowing algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. We propose a new linear system, which we refer to as the augmented normal equation ..."
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Cited by 20 (2 self)
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In this paper we develop a longstep primaldual infeasible pathfollowing algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. We propose a new linear system, which we refer to as the augmented normal equation (ANE), to determine the primaldual search directions. Since the condition number of the ANE coefficient matrix may become large for degenerate CQP problems, we use a maximum weight basis preconditioner introduced in [A. R. L. Oliveira and D. C. Sorensen, Linear
Probabilistic Analysis of Two Complexity Measures for Linear Programming Problems
 Math. Prog. A
, 1998
"... This note provides a probabilistic analysis of #A , a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A # IR mn . We show that if A is a standard Gaussian matrix, then E(ln #A ) = O(min{m ln n, n}). Thus, the e ..."
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Cited by 19 (3 self)
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This note provides a probabilistic analysis of #A , a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A # IR mn . We show that if A is a standard Gaussian matrix, then E(ln #A ) = O(min{m ln n, n}). Thus, the expected running time of linear programming is bounded by a polynomial in m and n for any righthand side and objective coefficient vectors when A is randomly generated in this way. We show that the same bound holds for E(ln #(A)), where #(A) is another condition number of A arising in complexity analyses of linear programming problems.
On the curvature of the central path of linear programming theory
 Foundations of Computational Mathematics
, 2003
"... Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming pro ..."
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Cited by 19 (3 self)
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Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming problem defined on a compact polytope contained in R n, the total curvature of the central path is less than or
A new complexity result on solving the Markov decision problem
 Mathematics of Operations Research
, 2005
"... We present a new complexity result on solving the Markov decision problem (MDP) with n states and a number of actions for each state, a special class of realnumber linear programs with the Leontief matrix structure. We prove that, when the discount factor θ is strictly less than 1, the pro ..."
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Cited by 14 (3 self)
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We present a new complexity result on solving the Markov decision problem (MDP) with n states and a number of actions for each state, a special class of realnumber linear programs with the Leontief matrix structure. We prove that, when the discount factor &theta; is strictly less than 1, the problem can be solved in at most O(n
On Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems
, 2003
"... In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of ..."
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Cited by 13 (8 self)
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In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of the distribution tails, we improve the existing moment estimates, and we prove various limit theorems for the cases where either n or m and n tend to in nity. Our results are of complexity theoretic interest, because interiorpoint methods and relaxation methods for the solution of Ax 0, x 6= 0 have running times that are bounded in terms of log C (A) and C (A) respectively. AMS Classi cation: primary 90C31,15A52; secondary 90C05,90C60,62H10. Key Words: condition number, random matrices, linear programming, probabilistic analysis, complexity theory.