We replace orthogonal projections in the Polyak subgradient method for nonnegatively constrained minimization with entropic projections, thus obtaining an interior-point subgradient method. Inexact entropic projection...
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We replace orthogonal projections in the Polyak subgradient method for nonnegatively constrained minimization with entropic projections, thus obtaining an interior-point subgradient method. Inexact entropic projections are quite cheap. Global convergence of the resulting method is established.
Three dynamical systems are associated with a problem of convex optimization in a finite-dimensional space. For system trajectories x(t), the ratios x(t)/t are, respectively, (i) solution tracking (staying within the ...
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Three dynamical systems are associated with a problem of convex optimization in a finite-dimensional space. For system trajectories x(t), the ratios x(t)/t are, respectively, (i) solution tracking (staying within the solution set X-0), (ii) solution abandoning (reaching X-0 as time t goes back to the initial instant), and (iii) solution approaching (approaching X-0 as time t goes to infinity). The systems represent a closed control system with appropriate feedbacks. In typical cases, the structure of the trajectories is simple enough. For instance, for a problem of quadratic programming with linear and box constraints, solution-approaching dynamics are described by a piecewise-linear ODE with a finite number of polyhedral domains of linearity. Finding the order of visiting these domains yields an analytic resolution of the original problem;a detailed analysis is given for a particular example. A discrete-time approach is outlined.
Given the minimization problem of a real-valued function f(x), x is an element of R-n, let A be any algorithm of type x(i+1) = x(i) + lambda(i)h(i), with lambda(i) is an element of R, h(i) is an element of R-n, -h(i)(...
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Given the minimization problem of a real-valued function f(x), x is an element of R-n, let A be any algorithm of type x(i+1) = x(i) + lambda(i)h(i), with lambda(i) is an element of R, h(i) is an element of R-n, -h(i)(T)del f(x(i)) greater than or equal to rho\\h(i)\\ \\del f (x(i))\\, rho is an element of (0, 1), that converges to a local minimum x* is an element of f(x). In this note, new assumptions on f(x) under which A converges linearly to x* are established. These include the ones introduced in the literature which involve the uniform convexity of f(x).
There has been considerable research in solving large-scale separable convex optimization problems. In this paper we present an algorithm for large-scale nonseparable smooth convex optimization problems with block-ang...
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There has been considerable research in solving large-scale separable convex optimization problems. In this paper we present an algorithm for large-scale nonseparable smooth convex optimization problems with block-angular linear constraints. The solution of the problem is approximated by solving a sequence of structured separable quadratic programs. The Bundle-based decomposition (BBD) method of Robinson (In: Prekopa, A., Szelezsan, J., Strazicky, B. (Eds.), System Modelling and Optimization, Springer, 1986, pp. 751-756;Annals de Institute Henri Poincare: Analyse Non Lineaire 6 (1989) 435-447) is applied to each separable quadratic program. We implement the algorithm and present computational experience. (C) 1998 Elsevier Science B.V. All rights reserved.
In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barri...
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In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see Yu. E. Nesterov and M.J. Todd, Math. Oper. Res., 22 (1997), pp. 1-42). The class of problems under consideration includes linear programming, semidefinite programming, and convex quadratically constrained, quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that typically go a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
Upper estimates are found for the sum of probabilities of all the events (x(1),...,x(r)), where xk is the frequency of the kth outcome in n independent trials carried out according to a polynomial scheme of trials wit...
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Upper estimates are found for the sum of probabilities of all the events (x(1),...,x(r)), where xk is the frequency of the kth outcome in n independent trials carried out according to a polynomial scheme of trials with r possible outcomes, the probability of each of which does not exceed the probability of a fixed event observed in n independent trials carried out according to the same scheme. Using these estimates we construct a test rejecting a polynomial scheme when the probabilities of outcomes in it are known.
We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the ...
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We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operator F, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures. (C) 1998 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
In the problem under consideration the given functionals of the trajectories must satisfy a system of inequalities. As usual, a construction of control strategy is required to ensure attainment of the extremal value o...
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In the problem under consideration the given functionals of the trajectories must satisfy a system of inequalities. As usual, a construction of control strategy is required to ensure attainment of the extremal value of the criterion. The paper provides an investigation of the space of strategy measures;necessary and sufficient optimality conditions are obtained, an algorithm of constructing optimal strategy for the Markovian case is developed, and an informative exactly solved example is given.
The theory of self-concordance in convex optimization has been used to analyze the complexity of interior-point methods based on Newton's method. For large problems, it may be impractical to use Newton's metho...
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The theory of self-concordance in convex optimization has been used to analyze the complexity of interior-point methods based on Newton's method. For large problems, it may be impractical to use Newton's method;here we analyze a truncated-Newton method, in which an approximation to the Newton search direction is used. In addition, practical interior-point methods often include enhancements such as extrapolation that are absent from the theoretical algorithms analyzed previously. We derive theoretical results that apply to such an algorithm, one similar to a sophisticated computer implementation of a barrier method. The results for a single barrier subproblem are a satisfying extension of the results for Newton's method. When extrapolation is used in the overall barrier method, however, our results are more limited. We indicate (by both theoretical arguments and examples) why more elaborate results may be difficult to obtain.
Primal-dual interior-point path-following methods for semidefinite programming are considered. Several variants are discussed, based on Newton's method applied to three equations: primal feasibility, dual feasibil...
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Primal-dual interior-point path-following methods for semidefinite programming are considered. Several variants are discussed, based on Newton's method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called the XZ, XZ+ZX, and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well defined and its Jacobian is nonsingular at the solution, under nondegeneracy assumptions. The associated Schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption. Practical aspects are discussed, including Mehrotra predictor-corrector variants and issues of numerical stability. Compared to the other methods considered, the XZ+ZX method is more robust with respect to its ability to step close to the boundary, converges more rapidly, and achieves higher accuracy.
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