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49
Game Theory, Maximum Entropy, Minimum Discrepancy And Robust Bayesian Decision Theory
 ANNALS OF STATISTICS
, 2004
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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 70 (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.
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 67 (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 expected total co ..."
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Cited by 20 (2 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.
Normal and Sinkless Petri Nets
 Journal of Computer and System Sciences
, 1989
"... We examine both the modeling power of normal and sinkless Petri nets and the computational complexities of various classical decision problems with respect to these two classes. We argue that although neither normal nor sinkless Petri nets are strictly more powerful than persistent Petri nets, th ..."
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Cited by 11 (5 self)
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We examine both the modeling power of normal and sinkless Petri nets and the computational complexities of various classical decision problems with respect to these two classes. We argue that although neither normal nor sinkless Petri nets are strictly more powerful than persistent Petri nets, they nonetheless are both capable of modeling a more interesting class of problems. On the other hand, we give strong evidence that normal and sinkless Petri nets are easier to analyze than persistent Petri nets. In so doing, we apply techniques originally developed for conflictfree Petri nets  a class defined solely in terms of the structure of the the net  to sinkless Petri nets  a class defined in terms of the behavior of the net. As a result, we give the first comprehensive complexity analysis of a class of potentially unbounded Petri nets defined in terms of their behavior. 1 Introduction Many aspects of the fundamental nature of computation are often studied via formal m...
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 9 (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...
Generation of basic semialgebraic invariants using convex polyhedra
 Static Analysis: Proceedings of the 12th International Symposium, volume 3672 of Lecture Notes in Computer Science
"... Abstract. A technique for generating invariant polynomial inequalities of bounded degree is presented using the abstract interpretation framework. It is based on overapproximating basic semialgebraic sets, i.e., sets defined by conjunctions of polynomial inequalities, by means of convex polyhedra. ..."
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Cited by 9 (0 self)
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Abstract. A technique for generating invariant polynomial inequalities of bounded degree is presented using the abstract interpretation framework. It is based on overapproximating basic semialgebraic sets, i.e., sets defined by conjunctions of polynomial inequalities, by means of convex polyhedra. While improving on the existing methods for generating invariant polynomial equalities, since polynomial inequalities are allowed in the guards of the transition system, the approach does not suffer from the prohibitive complexity of the methods based on quantifierelimination. The application of our implementation to benchmark programs shows that the method produces nontrivial invariants in reasonable time. In some cases the generated invariants are essential to verify safety properties that cannot be proved with classical linear invariants. 1
Establishing Wireless Conference Calls under Delay Constraints
 Journal of Algorithms
, 2002
"... A prevailing feature of mobile telephony systems is that the cell where a mobile user is located may be unknown. Therefore when the system is to establish a call between users it may need to search, or page, all the cells that it suspects the users are located in, to find the cells where the users c ..."
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Cited by 8 (4 self)
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A prevailing feature of mobile telephony systems is that the cell where a mobile user is located may be unknown. Therefore when the system is to establish a call between users it may need to search, or page, all the cells that it suspects the users are located in, to find the cells where the users currently reside. The search consumes expensive wireless links and so it is desirable to develop search techniques that page as few cells as possible.
Separating objects in the plane by wedges and strips
 Discrete Applied Mathematics
"... In this paper we study the separability of two disjoint sets of objects in the plane according to two criteria: wedge separability and strip separability. Wegivealgorithms for computing all the separating wedges and strips, the wedges with the maximum and minimum angle, and the narrowest and the wid ..."
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Cited by 6 (2 self)
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In this paper we study the separability of two disjoint sets of objects in the plane according to two criteria: wedge separability and strip separability. Wegivealgorithms for computing all the separating wedges and strips, the wedges with the maximum and minimum angle, and the narrowest and the widest strip. The objects we consider are points, segments, polygons and circles. As applications, we improve the computation of all the largest circles separating two sets of line segments by a log n factor, and we generalize the algorithm for computing the minimum polygonal separator of two sets of points to two sets of line segments with the same running time. Key words: Redblue separability, wedges, strips, circular and polygonal separability. 1