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70
ON THE RANK OF EXTREME MATRICES IN SEMIDEFINITE PROGRAMS AND THE MULTIPLICITY OF OPTIMAL EIGENVALUES
, 1998
"... We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalueoptimization, i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrixvalued function. We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically ..."
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Cited by 97 (1 self)
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We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalueoptimization, i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrixvalued function. We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalueoptimization. In the spectrum of an optimal matrix, the kth and (k / 1)st largest eigenvalues tend to be equal and frequently have multiplicity greater than two. This clustering is intuitively plausible and has been observed as early as 1975. When the matrixvalued function is affine, we prove that clustering must occur at extreme points of the set of optimal solutions, if the number of variables is sufficiently large. We also give a lower bound on the multiplicity of the critical eigenvalue. These results generalize to the case of a general matrixvalued function under appropriate conditions.
Game Theory, Maximum Entropy, Minimum Discrepancy And Robust Bayesian Decision Theory
 ANNALS OF STATISTICS
, 2004
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The nonlinear geometry of linear programming IV. Hilbert geometry, in preparation
"... This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope ..."
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Cited by 75 (0 self)
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This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure we study is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. In order to study Ptrajectories we also study a related vector field on the linear programming polytope, which we call the affine scaling vector field, and its associated trajectories, called Atrajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. These affine and projective scaling vector fields are each defined for liner programs of a special form, called strict standard form and canonical form, respectively. This paper defines and presents basic properties of Ptrajectories and Atrajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for Ptrajectories and Atrajectories. It presents Karmarkar’s interpretation of Atrajectories as steepest descent paths of the objective function 〈c, x 〉 with respect to the Riemannian _ dx
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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Stochastic Shortest Path Games
 SIAM Journal on Control and Optimization
, 1997
"... We consider dynamic, twoplayer, zerosum games where the #minimizing" player seeks to drive an underlying #nitestate dynamic system to a special terminal state along a least expected cost path. The #maximizer" seeks to interfere with the minimizer's progress so as to maximize the ex ..."
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Cited by 25 (3 self)
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We consider dynamic, twoplayer, zerosum games where the #minimizing" player seeks to drive an underlying #nitestate dynamic system to a special terminal state along a least expected cost path. The #maximizer" seeks to interfere with the minimizer's progress so as to maximize the expected total cost. We consider, for the #rst time, undiscounted #nitestate problems, with compact action spaces, and transition costs that are not strictly positive. We admit that there are policies for the minimizer which permit the maximizer to prolong the game inde#nitely. Under assumptions which generalize deterministic shortest path problems, we establish #i# the existence of a realvalued equilibrium cost vector achievable with stationary policies for the opposing players and #ii# the convergence of value iteration and policy iteration to the unique solution of Bellman's equation.
Sentential Probability Logic
, 1996
"... Introduction Among logicians it is wellknown that Leibniz was the first to conceive of a mathematical treatment of logic. Much less known, however, was his insistence that there was need for a new kind of logic that would treat of degrees of probability. Although it isn't clear what Leibniz ha ..."
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Cited by 23 (0 self)
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Introduction Among logicians it is wellknown that Leibniz was the first to conceive of a mathematical treatment of logic. Much less known, however, was his insistence that there was need for a new kind of logic that would treat of degrees of probability. Although it isn't clear what Leibniz had in mind for such a logic—understandably, since the subject of probability had just begun in his
Testing strictly concave rationality
 Journal of Economic Theory
, 1991
"... We prove that the Strong Axiom of Revealed Preference tests the existence of a strictly quasiconcave (in fact, continuous, generically Coo, strictly concave, and strictly monotone) utility function generating finitely many demand observations. This sharpens earlier results of Afriat, Diewert, and Va ..."
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Cited by 19 (1 self)
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We prove that the Strong Axiom of Revealed Preference tests the existence of a strictly quasiconcave (in fact, continuous, generically Coo, strictly concave, and strictly monotone) utility function generating finitely many demand observations. This sharpens earlier results of Afriat, Diewert, and Varian that tested ("nonparametrically") the existence of a piecewise linear utility function that could only weakly generate those demand observations. When observed demand is also invertible, we show that the rationalizing can be done in a Coo way, thus extending a result of Chiappori and Rochet from compact sets to all of R n • For finite data sets, one implication of our result is that even some weak types of rational behavior maximization of pseudotransitive or semitransitive preferences are observationally equivalent to maximization of continuous, strictly concave, and strictly monotone utility functions.
Switched networks and complementarity
 IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
"... Abstract—A modeling framework is proposed for circuits that are subject both to externally induced switches (time events) and to state events. The framework applies to switched networks with linear and piecewiselinear elements, including diodes. We show that the linear complementarity formulation, ..."
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Cited by 13 (7 self)
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Abstract—A modeling framework is proposed for circuits that are subject both to externally induced switches (time events) and to state events. The framework applies to switched networks with linear and piecewiselinear elements, including diodes. We show that the linear complementarity formulation, which already has proved effective for piecewiselinear networks, can be extended in a natural way to also cover switching circuits. To achieve this, we use a generalization of the linear complementarity problem known as the conecomplementarity problem. We show that the proposed framework is sound in the sense that existence and uniqueness of solutions is guaranteed under a passivity assumption. We prove that only firstorder impulses occur and characterize all situations that give rise to a state jump; moreover, we provide rules that determine the jump. Finally, we show that within our framework, energy cannot increase as a result of a jump, and we derive a stability result from this. Index Terms—Complementarity systems, hybrid systems, ideal diodes, ideal switches, piecewiselinear systems. I.
On the polyhedral decision problem
, 1977
"... Abstract. Computational problems sometimes can be cast in the following form: Given a point x in R, determine if x lies in some fixed polyhedron. In this paper we give a general lower bound to the complexity of such problems, showing that 1/2 log2 fs linear comparisons are needed in the worst case, ..."
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Cited by 12 (1 self)
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Abstract. Computational problems sometimes can be cast in the following form: Given a point x in R, determine if x lies in some fixed polyhedron. In this paper we give a general lower bound to the complexity of such problems, showing that 1/2 log2 fs linear comparisons are needed in the worst case, for any polyhedron with fs sdimensional faces. For polyhedra with abundant faces, this leads to lower bounds nonlinear in n, the number of variables.
The Theory of Linear Programming: Skew Symmetric SelfDual Problems and the Central Path
, 1994
"... The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Guler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple ..."
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Cited by 12 (6 self)
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The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Guler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple approach which is based on an embedding of the primal problem and its dual into a skew symmetric selfdual problem. This embedding is essentially due Ye, Todd and Mizuno. First we consider a skew symmetric selfdual linear program. We show that the strong duality theorem trivially holds in this case. Then, using the logarithmic barrier problem and the central path, the existence of a strictly complementary optimal solution is proved. Using the embedding just described, we easily obtain the strong duality theorem and the existence of strictly complementary optimal solutions for general linear programming problems. Key words. Linear programming, interior points, skew symmetric matrix, self...