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Optimal solutions to Markov decision problems may be very sensitive with respect to the state transition probabilities. In many practical problems, the estimation of these probabilities is far from accurate. Hence, estimation errors are limiting factors in applying Markov decision processes to real-world problems. We consider a robust control problem for a finite-state, finite-action Markov decision process, where uncertainty on the transition matrices is described in terms of possibly nonconvex sets. We show that perfect duality holds for this problem, and that as a consequence, it can be solved with a variant of the classical dynamic programming algorithm, the \"robust dynamic programming\" algorithm. We show that a particular choice of the uncertainty sets, involving likelihood regions or entropy bounds, leads to both a statistically accurate representation of uncertainty, and a complexity of the robust recursion that is almost the same as that of the classical recursion. Hence, robustness can be added at practically no extra computing cost. We derive similar results for other uncertainty sets, including one with a finite number of possible values for the transition matrices. We describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy; even if the uncertainty level is only crudely guessed, the robust strategy yields a much better worst-case expected travel time.

In this paper, we are concerned with a vector reverse convex minimization problem ( P ) . For such a problem, by means of the so-called Fenchel–Lagrange duality, we provide necessary optimality conditions for proper efficiency in the sense of Geoffrion. This duality is used after a decomposition of problem ( P ) into a family of convex vector minimization subproblems and scalarization of these subproblems. The optimality conditions are expressed in terms of subdifferentials and normal cones in the sense of convex analysis. The obtained results are new in the literature of vector reverse convex programming. Moreover, some of them extend with improvement some similar results given in the literature, from the scalar case to the vectorial one.

We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed.

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).In this paper we consider ... regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving ... regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the IHT method for finding an ...-local-optimal solution. We then propose a method for solving ... regularized convex cone programming by applying the IHT method to its quadratic penalty relaxation and establish its iteration complexity for finding an ...-approximate local minimizer. Finally, we propose a variant of this method in which the associated penalty parameter is dynamically updated, and show that every accumulation point is a local izer of the problem.

In this paper we consider l 0 regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving l 0 regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the IHT method for finding an ϵ -local-optimal solution. We then propose a method for solving l 0 regularized convex cone programming by applying the IHT method to its quadratic penalty relaxation and establish its iteration complexity for finding an ϵ -approximate local minimizer. Finally, we propose a variant of this method in which the associated penalty parameter is dynamically updated, and show that every accumulation point is a local izer of the problem.

In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss–Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an inexact sGS method, instead of an unknown number of cycles, to solve each of the subproblems involved. With some simple and implementable error tolerance criteria, the cost for solving the subproblems can be greatly reduced, and many steps in the forward sweep of each sGS cycle can often be skipped, which further contributes to the efficiency of the proposed method. Global convergence as well as the iteration complexity in the non-ergodic sense is established. Preliminary numerical experiments on some high-dimensional linear and convex quadratic SDP problems with a large number of linear equality and inequality constraints are also provided. The results show that for the vast majority of the tested problems, the sGS-imsPADMM is 2–3 times faster than the directly extended multi-block ADMM with the aggressive step-length of 1.618, which is currently the benchmark among first-order methods for solving multi-block linear and quadratic SDP problems though its convergence is not guaranteed.

Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. We extend Fourier-Motzkin elimination to semi-infinite linear programs, which are linear programs with finitely many variables and infinitely many constraints. Applying projection leads to new characterizations of important properties for primal-dual pairs of semi-infinite programs such as zero duality gap, feasibility, boundedness, and solvability. Extending the Fourier-Motzkin elimination procedure to semi-infinite linear programs yields a new classification of variables that is used to determine the existence of duality gaps. In particular, the existence of what the authors term dirty variables can lead to duality gaps. Our approach has interesting applications in finite-dimensional convex optimization. For example, sufficient conditions for a zero duality gap, such as the Slater constraint qualification, are reduced to guaranteeing that there are no dirty variables. This leads to completely new proofs of such sufficient conditions for zero duality.

One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformulation of the problem of interest and solving the resulting problem by a fast first order saddle point method utilizing smoothness of the saddle point cost function. In this paper, we demonstrate that when the saddle point cost function is polynomial, the precise gradients of the cost function required by deterministic first order saddle point algorithms and becoming prohibitively computationally expensive in the extremely large-scale case, can be replaced with incomparably cheaper computationally unbiased random estimates of the gradients. We show that for large-scale problems with favorable geometry, this randomization accelerates, progressively as the sizes of the problem grow, the solution process. This extends significantly previous results on acceleration by randomization, which, to the best of our knowledge, dealt solely with bilinear saddle point problems. We illustrate our theoretical findings by instructive and encouraging numerical experiments.

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend the ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of more than two separable convex functions. However, the convergence of this extension has been missing for a long time—neither an affirmative convergence proof nor an example showing its divergence is known in the literature. In this paper we give a negative answer to this long-standing open question: The direct extension of ADMM is not necessarily convergent. We present a sufficient condition to ensure the convergence of the direct extension of ADMM, and give an example to show its divergence.

We consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables. A Douglas-Rachford alternating direction method of multipliers (ADM) has been well studied in the literature for the special case of $m=2$. But the convergence of extending ADM to the general case of $m\\ge 3$ is still open. In this paper, we show that the straightforward extension of ADM is valid for the general case of $m\\ge 3$ if it is combined with a Gaussian back substitution procedure. The resulting ADM with Gaussian back substitution is a novel approach towards the extension of ADM from $m=2$ to $m\\ge 3$, and its algorithmic framework is new in the literature. For the ADM with Gaussian back substitution, we prove its convergence via the analytic framework of contractive-type methods, and we show its numerical efficiency by some application problems. [PUBLICATION ABSTRACT]