In this paper we study constraint qualifications and duality results for infinite convex programs (P) mu = inf{f(x): g(x) is-an-element-of -S, x is-an-element-of C}, where g = (g1, g2) and S = S1 x S2, S(i) are convex...
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In this paper we study constraint qualifications and duality results for infinite convex programs (P) mu = inf{f(x): g(x) is-an-element-of -S, x is-an-element-of C}, where g = (g1, g2) and S = S1 x S2, S(i) are convex cones, i = 1, 2, C is a convex subset of a vector space X, and f and g(i) are, respectively, convex and S(i)-convex, i = 1, 2. In particular, we consider the special case when S2 is in a finite dimensional space, g2 is affine and S2 is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case that g = g2 is affine and S = S2 is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption on g2 and the finite dimensionality assumption on S2, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.
This paper proposes a solution to the H-2 guaranteed cost control problem for uncertain, continuous-time linear systems. It consists of the determination of a constant state feedback stabilizing matrix gain and a H-2-...
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This paper proposes a solution to the H-2 guaranteed cost control problem for uncertain, continuous-time linear systems. It consists of the determination of a constant state feedback stabilizing matrix gain and a H-2-norm upper bound, valid for all feasible models. The uncertainties are only assumed to be convex-bounded, a concept which generalizes the important case of interval matrices uncertainties. The results follow from a new parameterization of all stabilizing gains over a convex set. As an additional property, the above mentioned H-2-norm upper bound reduces to the minimum H-2 cost in case of precisely known linear systems.
Based on the nearest-point projection of geometric convexity we give a general projection approach for solving the feasibility problem of linear programming. Application of Shor's method of space dilation gives ri...
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Based on the nearest-point projection of geometric convexity we give a general projection approach for solving the feasibility problem of linear programming. Application of Shor's method of space dilation gives rise to a family of polynomial-time ellipsoidal algorithms with improved termination criteria in case of infeasibility. Moreover, the approach renders possible application of various techniques from nonlinear programming. In particular, using a variable metric algorithm with exact line search we obtain a fast and practically well-behaving algorithm for linear programming.
We introduce a new algorithm for the continuous bounded quadratic knapsack problem. This algorithm is motivated by the geometry of the problem, is based on the iterative solution of a series of simple projection probl...
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We introduce a new algorithm for the continuous bounded quadratic knapsack problem. This algorithm is motivated by the geometry of the problem, is based on the iterative solution of a series of simple projection problems, and is easy to understand and implement. In practice, the method compares favorably to other well-known algorithms (some of which have superior worst-case complexity) on problem sizes up to n = 4000.
Within a Hilbert space we consider nonsmooth convex programs with sharp constraints. Examples include all problem instances that are bounded and strictly feasible. To solve such programs we pursue an absolutely contin...
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Within a Hilbert space we consider nonsmooth convex programs with sharp constraints. Examples include all problem instances that are bounded and strictly feasible. To solve such programs we pursue an absolutely continuous trajectory generated by a differential inclusion of subgradient type. Whenever this inclusion offers some freedom of choice, we select a steepest descent direction. It is shown that the proposed algorithm converges to an optimal solution in finite time.
The duality between maximum likelihood and entropy maximization problems is used to provide a primal-dual method for solving a maximum likelihood estimation problem arising from Positron Emission Tomography. We prove ...
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The duality between maximum likelihood and entropy maximization problems is used to provide a primal-dual method for solving a maximum likelihood estimation problem arising from Positron Emission Tomography. We prove convergence of our algorithm from the fact that the sequence it generates can be seen as the dual sequence produced by the hybrid version of Bregman's method when applied to a linearly constrained convex program with a Burg's entropy type objective function. This algorithm is shown to be closely connected to the so called Expectation Maximization (EM) algorithm.
To deal with the optimization problem min(x greater-than-or-equal-to 0)f(x), we propose a Gauss-Seidel type iterative approach where variables are modified sequentially one at a time (GSNA algorithm) or by blocks (BGS...
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To deal with the optimization problem min(x greater-than-or-equal-to 0)f(x), we propose a Gauss-Seidel type iterative approach where variables are modified sequentially one at a time (GSNA algorithm) or by blocks (BGSNA algorithm). Relying on both the Newton approach and an Armijo rule, an inaccurate line search is performed at each step to determine a step size insuring global convergence. Both robustness (global convergence) and efficiency (rate of local convergence) are analyzed under minimal hypotheses even weaker than those often required to prove the convergence of Gauss-Seidel methods. These procedures are applied to spatial price equilibrium problems, and the numerical results indicate that they are competitive with MINOS.
Minimum trace factor analysis is a commonly used technique for providing the greatest lower bound to reliability, and a modification of the basic problem involves the maximization of this greatest lower bound with res...
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Minimum trace factor analysis is a commonly used technique for providing the greatest lower bound to reliability, and a modification of the basic problem involves the maximization of this greatest lower bound with respect to suitably chosen weights. The underlying mathematical problems can be expressed as optimization problems with eigenvalue constraints, and it is well known that these can be nondifferentiable in the presence of multiple eigenvalues. In this paper, some recent developments in methods for working with constraints of this kind are exploited to provide methods which are second-order independent of the eigenvalue multiplicities. The effectiveness of the algorithms is demonstrated on some test problems.
This paper introduces two new proximal point algorithms for minimizing a proper, lower-semicontinuous convex function f: R-n -> R boolean OR {infinity}. Under this minimal assumption on f, the first algorithm posse...
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This paper introduces two new proximal point algorithms for minimizing a proper, lower-semicontinuous convex function f: R-n -> R boolean OR {infinity}. Under this minimal assumption on f, the first algorithm possesses the global convergence rate estimate f(x(k))-min(x is an element of Rn) f(x) = O(1/(Sigma(k-1)(j=0)root lambda(j))(2)), where {lambda(k)}(k=0)(infinity) are the proximal parameters. It is shown that this algorithm converges, and global convergence rate estimates for it are provided, even if minimizations are performed inexactly at each iteration. Both algorithms converge even if f has no minimizers or is unbounded from below. These algorithms and results are valid in infinite-dimensional Hilbert spaces.
Hiriart-Urruty gave formulas of the first-order and second-order ε-directional derivatives of a marginal function for a convex programming problem with linear equality constraints, that is, the image of a function un...
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Hiriart-Urruty gave formulas of the first-order and second-order ε-directional derivatives of a marginal function for a convex programming problem with linear equality constraints, that is, the image of a function under linear mapping (Ref. 1). In this paper, we extend his results to a problem with linear inequality constraints. The formula of the first-order derivative is given with the help of a duality theorem. A lower estimate for the second-order ε-directional derivative is given.
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