In this work, we study discrete-time Markov decision processes (MDPs) under constraints with Borel state and action spaces and where all the performance functions have the same form of the expected total reward (ETR) ...
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In this work, we study discrete-time Markov decision processes (MDPs) under constraints with Borel state and action spaces and where all the performance functions have the same form of the expected total reward (ETR) criterion over the infinite time horizon. One of our objective is to propose a convex programming formulation for this type of MDP. It will be shown that the values of the constrained control problem and the associated convex program coincide. Moreover, if there exists an optimal solution to the convex program then there exists a stationary randomized policy which is optimal for the MDP. It will be also shown that in the framework of constrained control problems, the supremum of the ETRs over the set of randomized policies is equal to the supremum of the ETRs over the set of stationary randomized policies. We consider standard hypotheses such as the so-called continuity-compactness conditions and a Slater-type condition. Our assumptions are quite weak to deal with cases that have not yet been addressed in the literature. Examples are presented to illustrate our results.
A differential inclusion is designed for solving cone-constrained convex programs. The method is of subgradient-projection type. It involves projection, penalties and Lagrangian relaxation. Non-smooth data can be acco...
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A differential inclusion is designed for solving cone-constrained convex programs. The method is of subgradient-projection type. It involves projection, penalties and Lagrangian relaxation. Non-smooth data can be accommodated. A novelty is that multipliers converge monotonically upwards to equilibrium levels. An application to stochastic programming is considered.
The Douglas-Rachford splitting method is a classical and powerful method for minimizing the sum of two convex functions. In this paper, we introduce two dynamical systems based on this method for solving the minimum v...
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The Douglas-Rachford splitting method is a classical and powerful method for minimizing the sum of two convex functions. In this paper, we introduce two dynamical systems based on this method for solving the minimum value problem of the sum of a strongly convex function and a weakly convex function. Under mild conditions, it is shown that the proposed dynamical systems are globally convergent to fixed point sets of the corresponding Douglas-Rachford operators, respectively, and are globally asymptotically stable if the corresponding fixed point sets are singleton. Furthermore, the globally exponential convergence of the proposed dynamical systems is established under some regularity conditions. A numerical example is reported to illustrate the effectiveness of the dynamical splitting method.
This paper presents a new class of outer approximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. ...
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This paper presents a new class of outer approximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. Moreover, the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem, while other outer approximation methods usually use the original objective function itself throughout the iterations. By this modification, convergence of the methods can be proved under mild conditions. Furthermore, it is shown that generalized versions of the cut construction schemes in Kelley-Cheney-Goldstein's cutting plane method and Veinott's supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations.
A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the o...
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A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the objective function can be described by a continuously differentiable convex *** a first part, a cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and, if not, imposes a convexity cut which rejects the whole cell from the feasibility region. This elimination, which is basically a dual decomposition method but with an efficient use of the specific structure of the problem is shown to be finitely *** second part of this paper is devoted to the study of some special cases of piecewise convex program and in particular the piecewise quadratic program having a polyhedral constraint set. Such a program arises naturally in stochastic quadratic programming with recourse, which is the subject of the last section.
We introduce and characterize a class of differentiable convex functions for which the Karush-Kuhn-Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterizati...
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We introduce and characterize a class of differentiable convex functions for which the Karush-Kuhn-Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterization of optimality generally assumes an asymptotic form. We also show that for the functions that belong to this class in multi-objective optimization, Pareto solutions coincide with strong Pareto solutions,. This extends a result, well known for the linear case.
A variant of the embedding technique proposed earlier by the second author is suggested in which the sets to be embedded are support cones. Replacing the cones by simplices gives a modification with a polynomial conve...
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A variant of the embedding technique proposed earlier by the second author is suggested in which the sets to be embedded are support cones. Replacing the cones by simplices gives a modification with a polynomial convergence rate.
The authors present a primal interior-point algorithm for solving convex programs with nonlinear constraints. The algorithm uses a predictor-corrector strategy to follow a smooth path that leads from a given starting ...
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The authors present a primal interior-point algorithm for solving convex programs with nonlinear constraints. The algorithm uses a predictor-corrector strategy to follow a smooth path that leads from a given starting point to an optimal solution. A convergence analysis is given showing that under mild assumptions the algorithm simultaneously iterates towards feasibility and optimality. The matrices involved can be kept sparse if the nonlinear functions are separable or depend on only a few variables. A preliminary implementation has been developed. Some promising numerical results indicate that the algorithm may be efficient in practice, and that it can deal in a single phase with infeasible starting points without relying on some ''big M'' parameter.
A parallel algorithm is proposed in this paper for solving the problem $\min \{ q(x)|x \in C_1 \cap \cdots \cap C_m \} $ where q is an uniformly convex function and $C_i$ are closed convex sets in $R^n$. In each it...
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A parallel algorithm is proposed in this paper for solving the problem $\min \{ q(x)|x \in C_1 \cap \cdots \cap C_m \} $ where q is an uniformly convex function and $C_i$ are closed convex sets in $R^n$. In each iteration of the method, we solve in parallel m independent subproblems, each minimizing a definite quadratic function over an individual set $C_i$. The method has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs, linear programs, systems of linear equations and systems of generalized nonlinear inequalities.
In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such...
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In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of low-spikiness in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the joint estimation provides a significant gain over the individual recovery of each matrix when the number of matrices sharing a column space and the ambient dimension of the shared column space are large relative to the number of columns in each matrix. The convex estimator is cast as a semidefinite program and an efficient ADMM algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.
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