Ellipsoid Method

We consider the problem of online convex optimization against an arbitrary adversary with bandit feedback, known as bandit convex optimization. We give the first Õ( √ T )-regret algorithm for this setting based on a novel application of the ellipsoid method to online learning. This bound is known to be tight up to logarithmic factors. Our analysis introduces new tools in discrete convex geometry.

We consider the problem of online convex optimization against an arbitrary adversary with bandit feedback, known as bandit convex optimization. We give the first Õ( √ T )-regret algorithm for this setting based on a novel application of the ellipsoid method to online learning. This bound is known to be tight up to logarithmic factors. Our analysis introduces new tools in discrete convex geometry.

The research reported in this paper is concerned with an application of the ellipsoid algorithm in the interactive multieriteria linear programming step method (STEM) by Benayoun et al. [1971]. Due to this application we eliminate some drawbacks of the original version of STEM and, moreover, we avoid extra computations connected with sensitivity analysis in every iteration. Specifically, we use the ellipsoid algorithm to minimize the Euclidean norm in the criterion space instead of the Chebyshev norm, which ensures that every solution submitted to the decision maker is efficient. As follows from a computational experiment, in comparison with the application of the simplex method, the proposed modification of STEM shows a smaller increase of the computational effort when the number of criteria increases. However, the absolute computation time becomes worse for problems of larger size. Zusammenfassung: In dieser Arbeit wird fiber eine Anwendungsm6glichkeit der Khachiyan-Shor-Algorithmus (EUipsoid-Algorithmus) im Rahmen des STEM-Verfahrens zur interaktiven L6sung linearer VektoroptimierungsmodeUe beriehtet. Auf diese Weise k6nnen einige spezifisehe Nachteile des STEM-Verfahrens in seiner Originalversion vermieden werden. Durch die Verwendung der Euklidischen Norm anstelle der beim STEM-Verfahren iiblichen Tschebyscheff-Norm wird garantiert, daft dem Entscheidungstr~iger nur effiziente L6sungen vorgeschlagen werden. Die numerischen Erfahrungen zeigen, daft der L6sungsaufwand der hier vorgeschlagenen Modifikation des STEM-Verfahrens mit steigender Anzahl yon Zielfunktionen weniger stark zunimmt als bei der fiblichen Version. Dies gilt jedoch nicht hinsichtlich der allgemeinen Problemgr6t~e:

A loop-shaping approach for tuning Proportional-Integral-Derivative (PID) controllers is presented. A Glover-McFarlane controller is used to determine a target loop-shape that is approximated by a PID structure via the use of an LMI optimization method. If the approximation error from the minimization satisfies a standard small gain condition, then the tuned PID controller also shares the attractive robustness properties of the controllers. The entire design process is carried out in discrete-time assuming a discrete plant is available. Typical test cases are used to show the implementation of this method and the quality of the resulting closed loops.

In this paper, Banach space duality theory for the multiobjective H 2 / H m problem developed recently by the authors, is used to develop algorithms to solve this problem by approximately reducing the dual and predual reperesentations to finite variable convex optimizations. The Ellipsoid algorithm is then a p plied to these problems to obtain polynomial-time, nonheuristic programs which find "nearly" optimal control laws. These algorithms have been implemented numerically to compute an example.

We introduce a new type of agent whom we refer to as a "problem solver" (PS). The PS interacts with conventional players and wishes to respond optimally to their moves. The PS has only partial information about the moves of the other players. Unlike a regular player, the PS does not "put himself in the shoes" of other players and does not form beliefs about their moves. Rather, he treats the data as a logic puzzle: he calculates the set of possible configurations of moves that are consistent with the data he observes and responds to it. We insert such a problem solver into a simple model of competition for attention and analyze its equilibria. We demonstrate a novel feature of equilibrium in the model, whereby even though the PS always succeeds in his task, he may be uncertain that he will do so. The second author acknowledges financial support from ERC grant 269143. Our thanks to Noga Alon, Ayala Arad, Kfir Eliaz, Gil Kalai and Rani Spiegler for helpful conversations.

This paper proposes a novel ellipsoid method for the optimization of electromagnetic constrained problems. Unlike the classical method, which can apply only one cut per iteration, this novel algorithm can employ multiple cuts simultaneously. This improves the convergence rate while preserving all theoretical guarantees of the original method. The design of modelled reflector antennas for optimal coverage of the Brazilian, Chinese, and American territories is presented. These problems have 38 control variables and numerous constraints (18, 11, and 12). Several results are presented for the enhanced and classical ellipsoid methods, as well as for stochastic algorithms. They assert the efficiency of the introduced technique.

The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables [Formula: see text] when its set of solutions has positive volume. However, when [Formula: see text] is infeasible, the ellipsoid algorithm has no mechanism for proving that (P) is infeasible. This is in contrast to the other two fundamental algorithms for tackling [Formula: see text], namely, the simplex and interior-point methods, each of which can be easily implemented in a way that either produces a solution of [Formula: see text] or proves that [Formula: see text] is infeasible by producing a solution to the alternative system [Formula: see text]. This paper develops an oblivious ellipsoid algorithm (OEA) that either produces a solution of [Formula: see text] or produces a solution of [Formula: see text]. Depending on the dimensions and other condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better t...

Given A := {a 1 , . . . , a m } ⊂ R d whose affine hull is R d , we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a new algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yıldırım, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotical complexity bound of the algorithm of Kumar and Yıldırım or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.

Given a set of points S = {x 1 ,. .. , x m } ⊂ R n and > 0, we propose and analyze an algorithm for the problem of computing a (1 +)-approximation to the the minimum volume axis-aligned ellipsoid enclosing S. We establish that our algorithm is polynomial for fixed. In addition, the algorithm returns a small core set X ⊆ S, whose size is independent of the number of points m, with the property that the minimum volume axis-aligned ellipsoid enclosing X is a good approximation of the minimum volume axis-aligned ellipsoid enclosing S. Our computational results indicate that the algorithm exhibits significantly better performance than that indicated by the theoretical worst-case complexity result.

Given A := {a 1 ,. .. , a m } ⊂ R d whose affine hull is R d , we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a new algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yıldırım, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotical complexity bound of the algorithm of Kumar and Yıldırım or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.

This paper establishes IQC-based (Integral Quadratic Constraints) conditions under which an ellipsoid is contractively invariant for a single input linear system under a saturated linear feedback law. Based on these set invariance conditions, the determination of the largest such ellipsoid, for use as an estimate of the domain of attraction, can be formulated and solved as an LMI optimization problem. Such an LMI problem can also be readily adapted for the design of the feedback gain that achieves the largest contractively invariant ellipsoid. While the advantages of the proposed IQC approach remain to be explored, it is shown in this paper that the largest contractively invariant ellipsoid determined by this approach is the same as the one determined by the existing approach based on expressing the saturated linear feedback as a linear differential inclusion (LDI), which is known to lead to non-conservative result in determining the largest contractively invariant ellipsoid for single input systems.

A bs 1 ra c t : An ellipsoid algorithm for solving nonlinear programming problems is described, whose objective and constraints functions are convex and are not assumed to be differentiable. The original ellipsoid algorithm was suggested by ludin and Nemirovskii and used by Shor for solving convex nonlinear programming problems. This paper suggests some modifications to the basic ellipsoid algorithm to adapt it to magnetostatic applications. One way to do this is to control the new ellipsoid at each iteration to generate smaller ellipsoids and to speedup the basic algorithm. Finally, illustrative examples show the applicability of the method to solving constrained nonlinear optimization problems in magnetostatic.

CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) 3 Conclusion: Manually adjusted MRI T2-weighted segmentation is likely the most accurate measure of total prostate volume. DynaCAD appears to fulfill that function, but manual adjustment of automatic misregistration of boundaries is necessary. ACAD and RCAD are best applied to research use. Ellipsoid methods are faster, more convenient, nearly as accurate and more practical for clinical use.

This paper presents a novel repeater insertion algorithm for the power minimization of realistic interconnect trees under given timing budgets. Our algorithm judiciously combines a local optimizer based on the dynamic programming technique and a global search engine using the ellipsoid method. As a result, our approach is capable of producing high-quality solutions at a very fast speed. Furthermore, our scheme is robust and does not need any manual tuning of the iteration-control parameters. We have developed a repeater insertion tool, called FREEZE, using the proposed algorithm and applied it to various interconnect trees with different timing targets. Experimental results demonstrate the high effectiveness of our approach. In comparison with the state-of-the-art low-power repeater insertion schemes, FREEZE requires 5.8 times fewer iterations on the average, achieving up to 27 times speedup with even better power savings. When compared with a dynamic programming based scheme, which guarantees the optimal solution, our tool delivers up to 50 times speedup with 0.9% power increase on the average.

In this paper we show the rst polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds of nite size. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap [D. Gijswijt and G. Pap, An algorithm for weighted fractional matroid matching, J. Combin. Theory, Ser. B 103 (2013), 509520].

The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with "small" subdeterminants, and the polynomiality of such integer problems, provided the integer linear version of such problems is polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems with or without integer variables, to procedures for solving the respective nonlinear separable problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming problems. An algorithm for finding an c-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(l/c) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function. The practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function, only a polynomial number of function evaluations on a prespecified grid.

In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for minimizing an indefinite quadratic function over a bounded polyhedral convex set which is not necessarily given explicitly by a system of linear inequalities and/or equalities. It is required that for this set there exists an efficient algorithm to verify whether a point is feasible, and

M-convex functions have various desirable properties as convexity in discrete optimization. We can find aglobal minimum of an M-convex function by agreedy algorithm, i.e., s0-called descent algorithms work for the minimization. In this paper, we apply ascaling technique to agreedy algorithm and propose an efficient algorithm for the minimization of an $\mathrm{M}$-convex function. Computational results are also reported.

A new technique for design centering, and for polytope approximation of the feasible region for a design are presented. In the rst phase, the feasible region is approximated by a convex polytope, using a method based on a theorem on convex sets. As a natural consequence of this approach, a good approximation to the design center is obtained. In the next phase, the exact design center is estimated using one of two techniques that we present in this paper. The rst inscribes the largest Hessian ellipsoid, which is known to be a good approximation to the shape of the polytope, within the polytope. This represents an improvement o ver previous methods, such as simplicial approximation, where a hypersphere or a crudely estimated ellipsoid is inscribed within the approximating polytope. However, when the pdf's of the design parameters are known, the design center does not necessarily correspond to the center of the largest inscribed ellipsoid. Hence, a second technique is developed, which incorporates the probability distributions of the parameters, under the assumption that their variation is modeled by Gaussian probability distributions. The problem is formulated as a convex programming problem and an e cient algorithm is used to calculate the design center, using fast and e cient Monte Carlo methods to estimate the yield gradient. An example is provided to illustrate how ellipsoid-based methods fail to incorporate the probability density functions, and is solved using the convex programming-based algorithm.

A new exact approach to the stable set problem is presented, which attempts to avoid the pitfalls of existing approaches based on linear and semidefinite programming. The method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimising over it is equal to the Lovász theta number. This ellipsoid can then be used to construct useful convex relaxations of the stable set problem, which can be embedded within a branch-and-bound framework. Extensive computational results are given, which indicate the potential of the approach.

In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of the stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented. Key Words. Location theory, min-max programming, ellipsoid algorithm, quasiconvexity. ~The work of the second author was supported by JNICT (Portugal), under Contract BD/ 631/90-RM, during his stay at Erasmus University in Rotterdam. 2The authors would like to thank the anonymous referees for simplifying the proofs in the first part of Section 2 and for their constructive remarks improving the presentation.

A new technique for design centering, and for polytope approximation of the feasible region for a design are presented. In the rst phase, the feasible region is approximated by a convex polytope, using a method based on a theorem on convex sets. As a natural consequence of this approach, a good approximation to the design center is obtained. In the next phase, the exact design center is estimated using one of two techniques that we present in this paper. The rst inscribes the largest Hessian ellipsoid, which is known to be a good approximation to the shape of the polytope, within the polytope. This represents an improvement o ver previous methods, such as simplicial approximation, where a hypersphere or a crudely estimated ellipsoid is inscribed within the approximating polytope. However, when the pdf's of the design parameters are known, the design center does not necessarily correspond to the center of the largest inscribed ellipsoid. Hence, a second technique is developed, which incorporates the probability distributions of the parameters, under the assumption that their variation is modeled by Gaussian probability distributions. The problem is formulated as a convex programming problem and an e cient algorithm is used to calculate the design center, using fast and e cient Monte Carlo methods to estimate the yield gradient. An example is provided to illustrate how ellipsoid-based methods fail to incorporate the probability density functions, and is solved using the convex programming-based algorithm.

A loop-shaping approach for tuning Proportional-Integral-Derivative (PID) controllers is presented. A Glover-McFarlane controller is used to determine a target loop-shape that is approximated by a PID structure via the use of an LMI optimization method. If the approximation error from the minimization satisfies a standard small gain condition, then the tuned PID controller also shares the attractive robustness properties of the controllers. The entire design process is carried out in discrete-time assuming a discrete plant is available. Typical test cases are used to show the implementation of this method and the quality of the resulting closed loops.

Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of recent advances in equilibrium computation for probabilistic inference. We present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. We concretely establishes the precise connection between variational probabilistic inference in MRFs and correlated equilibria. No previous work exploits recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. We discuss how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. Also, inspired by a previously...

In autosomal dominant polycystic kidney disease (ADPKD), total kidney volume (TKV) is regarded as an important biomarker of disease progression and different methods are available to assess kidney volume. The purpose of this study was to identify the most efficient kidney volume computation method to be used in clinical studies evaluating the effectiveness of treatments on ADPKD progression. We measured single kidney volume (SKV) on two series of MR and CT images from clinical studies on ADPKD (experimental dataset) by two independent operators (expert and beginner), twice, using all of the available methods: polyline manual tracing (reference method), free-hand manual tracing, semi-automatic tracing, Stereology, Mid-slice and Ellipsoid method. Additionally, the expert operator also measured the kidney length. We compared different methods for reproducibility, accuracy, precision, and time required. In addition, we performed a validation study to evaluate the sensitivity of these me...

Strategic modelling with a panoramic view plays an important role in decision-making problems. It offers the possibility of generating different solutions before making a decision. This is particularly relevant in critical situations. This article addresses the problem of allocating resources, whether financial, material or human, so that it is optimal under a given set of constraints and inter-dependencies with other systems. To do this, existing strategies such as those of Colonel Blotto are studied in order to evaluate them according to some criteria, including the heterogeneity or homogeneity of resources and/or battlefields. Based on the results of these configurations, we propose distributed strategic learning methods to find better resource allocation strategies. The proposed algorithms are implemented under various scenarios, including incomplete information. Case studies are carried out to test the effectiveness of these new strategies compared to previous ones. A complexit...

Strategic modelling with a panoramic view plays an important role in decision-making problems. It offers the possibility of generating different solutions before making a decision. This is particularly relevant in critical situations. This article addresses the problem of allocating resources, whether financial, material or human, so that it is optimal under a given set of constraints and inter-dependencies with other systems. To do this, existing strategies such as those of Colonel Blotto are studied in order to evaluate them according to some criteria, including the heterogeneity or homogeneity of resources and/or battlefields. Based on the results of these configurations, we propose distributed strategic learning methods to find better resource allocation strategies. The proposed algorithms are implemented under various scenarios, including incomplete information. Case studies are carried out to test the effectiveness of these new strategies compared to previous ones. A complexit...

The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with “small” subdeterminants, and the polynomiality of such integer problems, provided the inteter linear version of such problems ins polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming. An algorithm for finding a ε-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(1/ε) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and...

The extended Kalman filter (EKF) simultaneous localization and mapping (SLAM) requires the uncertainty to be Gaussian noise. This assumption can be relaxed to bounded noise by the set membership SLAM. However, the published set membership SLAMs are not suitable for largescale and online problems. In this paper, we use ellipsoid algorithm for solving SLAM problem. The proposed ellipsoid SLAM has advantages over EKF SLAM and the other set membership SLAMs, in noise condition, online realization, and large-scale problem. By bounded ellipsoid technique, we analyze the convergence and stability of the ellipsoid SLAM. Simulation and experimental results show that the proposed ellipsoid SLAM is effective for online and large-scale problems such as Victoria Park dataset.

In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simultaneously distribute their troops across different battlefields.The winner of each battlefield is determined independently by a winner-take-all rule. The ultimate payoff of each colonel is the number of battlefields he wins. This game is commonly used for analyzing a wide range of applications such as the U.S presidential election, innovative technology competitions, advertisements, etc. There have been persistent efforts for finding the optimal strategies for the Colonel Blotto game. After almost a century Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin provided a poly-time algorithm for finding the optimal strategies. They first model the problem by a Linear Program (LP) with exponential number of constraints and use Ellipsoid method to solve it. However, despite the theoretical importance of their algorithm, it ishighly impractical. In general, even Simplex method (despite it...

Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of recent advances in equilibrium computation for probabilistic inference. We present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. We concretely establishes the precise connection between variational probabilistic inference in MRFs and correlated equilibria. No previous work exploits recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. We discuss how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. Also, inspired by a previously...

This paper presents a novel repeater insertion algorithm for the power minimization of realistic interconnect trees under given timing budgets. Our algorithm judiciously combines a local optimizer based on the dynamic programming technique and a global search engine using the ellipsoid method. As a result, our approach is capable of producing high-quality solutions at a very fast speed. Furthermore, our scheme is robust and does not need any manual tuning of the iteration-control parameters. We have developed a repeater insertion tool, called FREEZE, using the proposed algorithm and applied it to various interconnect trees with different timing targets. Experimental results demonstrate the high effectiveness of our approach. In comparison with the state-of-the-art low-power repeater insertion schemes, FREEZE requires 5.8 times fewer iterations on the average, achieving up to 27 times speedup with even better power savings. When compared with a dynamic programming based scheme, which guarantees the optimal solution, our tool delivers up to 50 times speedup with 0.9% power increase on the average.

Given A := {a 1 ,. .. , a m } ⊂ R d whose affine hull is R d , we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a new algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yıldırım, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotical complexity bound of the algorithm of Kumar and Yıldırım or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.

We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players.

We propose a column-eliminating technique for the simplex method of linear programming. A pricing criterion is developed for checking whether a dual hyperplane corresponding to a column intersects a simplex containing all of the optimal dual feasible solutions. If the dual hyperplane has no intersection with this simplex, we can eliminate the corresponding column from further computation during the course of the simplex method.

The conjugate gradient (CG) algorithm is well-known to have excellent theoretical properties for solving linear systems of equations Ax = b where the n × n matrix A is symmetric positive definite. However, for extremely ill-conditioned matrices the CG algorithm performs poorly ...

Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of recent advances in equilibrium computation for probabilistic inference. We present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. We concretely establishes the precise connection between variational probabilistic inference in MRFs and correlated equilibria. No previous work exploits recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. We discuss how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. Also, inspired by a previously...

The conjugate gradient (CG) algorithm is well-known to have excellent theoretical properties for solving linear systems of equations Ax = b where the n £ n matrix A is symmetric positive definite. However, for extremely ill-conditioned matrices the CG algorithm performs poorly in practice. In this paper, we discuss an adaptive preconditioning procedure which improves the performance of the CG algorithm on extremely ill-conditioned systems. We introduce the preconditioning procedure by applying it first to the steepest descent algorithm. Then, the same techniques are extended to the CG algorithm, and convergence to an †-solution in O(logdet(A) + p nlog† i1 ) iterations is proven, where det(A) is the determinant of the matrix.

The conjugate gradient (CG) algorithm is well-known to have excellent theoretical properties for solving linear systems of equations Ax = b where the n × n matrix A is symmetric positive definite. However, for extremely ill-conditioned matrices the CG algorithm performs poorly in practice. In this paper, we discuss an adaptive preconditioning procedure which improves the performance of the CG algorithm on extremely ill-conditioned systems. We introduce the preconditioning procedure by applying it first to the steepest descent algorithm. Then, the same techniques are extended to the CG algorithm, and convergence to an-solution in O(log det(A) + √ n log −1) iterations is proven, where det(A) is the determinant of the matrix.

We propose a column-eliminating technique for the simplex method of linear programming. A pricing criterion is developed for checking whether a dual hyperplane corresponding to a column intersects a simplex containing all of the optimal dual feasible solutions. If the dual hyperplane has no intersection with this simplex, we can eliminate the corresponding column from further computation during the course of the simplex method.

This paper introduces a finite-time algorithm which allows a hybrid system t o determine whether or not a specified symbolic behaviour can be realized by the system. The proposed algorithm is an inductive inference protocol based on the ellipsoid method. It can be viewed as a means by which the hybrid system can autonomously set its achievable goal behaviour. This ability for goal self-determination is argued as an important attribute of intelligent control systems. 'The partial financial support of the National Science Foundation (IRI91-09298 and MSS92-16559) is gratefully acknowledged.

... G. Sometimes partial inequality systems turn out to be quite helpful for solving special eases. ... V,E] which satisfies polyhedron determined by the trivial inequalities (3.1) and the odd cycle constraints ... path problem in G can be solved by calculating a minimum perfect matching in ...

We analyze the problem of computing a correlated equilibrium that optimizes some objective (e.g., social welfare). Papadimitriou and Roughgarden [2008] gave a sufficient condition for the tractability of this problem; however, this condition only applies to a subset of existing representations. We propose a different algorithmic approach for the optimal CE problem that applies to all compact representations, and give a sufficient condition that generalizes that of Papadimitriou and Roughgarden [2008]. In particular, we reduce the optimal CE problem to the deviation-adjusted social welfare problem, a combinatorial optimization problem closely related to the optimal social welfare problem. This framework allows us to identify new classes of games for which the optimal CE problem is tractable; we show that graphical polymatrix games on tree graphs are one example. We also study the problem of computing the optimal coarse correlated equilibrium, a solution concept closely related to CE. Using a similar approach we derive a sufficient condition for this problem, and use it to prove that the problem is tractable for singleton congestion games.

In a landmark paper, Papadimitriou and Roughgarden [2008] described a polynomial-time algorithm ("Ellipsoid Against Hope") for computing sample correlated equilibria of concisely-represented games. Recently, Stein, Parrilo and Ozdaglar [2010] showed that this algorithm can fail to find an exact correlated equilibrium, but can be easily modified to efficiently compute approximate correlated equilibria. Currently, it remains an open problem to determine whether the algorithm can be modified to compute an exact correlated equilibrium. We show that it can, presenting a variant of the Ellipsoid Against Hope algorithm that guarantees the polynomial-time identification of exact correlated equilibrium. Also, our algorithm is the first to tractably compute correlated equilibria with polynomial-sized supports; such correlated equilibria are more natural solutions than the mixtures of product distributions produced previously, and have several advantages including requiring fewer bits to represent, being easier to sample from, and being easier to verify. Our algorithm differs from the original primarily in its use of a separation oracle that produces cuts corresponding to pure-strategy profiles. As a result, we no longer face the numerical precision issues encountered by the original approach, and both the resulting algorithm and its analysis are considerably simplified. Our new separation oracle can be understood as a derandomization of Papadimitriou and Roughgarden's original separation oracle via the method of conditional probabilities. We also adapt our techniques to two related algorithms that are based on the Ellipsoid Against Hope approach, Hart and Mansour's [2010] CE procedure with polynomial communication complexity and Huang and Von Stengel's [2008] polynomial-time algorithm for extensive-form correlated equilibria, yielding in both cases exact solutions with polynomial-sized supports.

We analyze the problem of computing a correlated equilibrium that optimizes some objective (e.g., social welfare). Papadimitriou and Roughgarden [2008] gave a sufficient condition for the tractability of this problem; however, this condition only applies to a subset of existing representations. We propose a different algorithmic approach for the optimal CE problem that applies to all compact representations, and give a sufficient condition that generalizes that of Papadimitriou and Roughgarden [2008]. In particular, we reduce the optimal CE problem to the deviation-adjusted social welfare problem, a combinatorial optimization problem closely related to the optimal social welfare problem. This framework allows us to identify new classes of games for which the optimal CE problem is tractable; we show that graphical polymatrix games on tree graphs are one example. We also study the problem of computing the optimal coarse correlated equilibrium, a solution concept closely related to CE. Using a similar approach we derive a sufficient condition for this problem, and use it to prove that the problem is tractable for singleton congestion games.

In a landmark paper, Papadimitriou and Roughgarden [2008] described a polynomial-time algorithm ("Ellipsoid Against Hope") for computing sample correlated equilibria of concisely-represented games. Recently, Stein, Parrilo and Ozdaglar [2010] showed that this algorithm can fail to find an exact correlated equilibrium, but can be easily modified to efficiently compute approximate correlated equilibria. Currently, it remains an open problem to determine whether the algorithm can be modified to compute an exact correlated equilibrium. We show that it can, presenting a variant of the Ellipsoid Against Hope algorithm that guarantees the polynomial-time identification of exact correlated equilibrium. Also, our algorithm is the first to tractably compute correlated equilibria with polynomial-sized supports; such correlated equilibria are more natural solutions than the mixtures of product distributions produced previously, and have several advantages including requiring fewer bits to represent, being easier to sample from, and being easier to verify. Our algorithm differs from the original primarily in its use of a separation oracle that produces cuts corresponding to pure-strategy profiles. As a result, we no longer face the numerical precision issues encountered by the original approach, and both the resulting algorithm and its analysis are considerably simplified. Our new separation oracle can be understood as a derandomization of Papadimitriou and Roughgarden's original separation oracle via the method of conditional probabilities. We also adapt our techniques to two related algorithms that are based on the Ellipsoid Against Hope approach, Hart and Mansour's [2010] CE procedure with polynomial communication complexity and Huang and Von Stengel's [2008] polynomial-time algorithm for extensive-form correlated equilibria, yielding in both cases exact solutions with polynomial-sized supports.