The Nesterov-Todd Direction and its Relation to Weighted Analytic Centers

The subject of this report concerns differential-geometric properties of the Nesterov-Todd search direction for linear optimization over symmetric cones. In particular, we investigate the rescaled asymptotics of the associated flow near the central path. Our results imply that the Nesterov-Todd direction arises as the solution of a Newton system defined in terms of a certain transformation of the primal-dual feasible domain. This transformation has especially appealing properties which generalize the notion of weighted analytic centers for linear programming

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

The Stochastic Shortest Path Problem : A polyhedral combinatorics perspective

In this paper, we give a new framework for the stochastic shortest path problem in finite state and action spaces. Our framework generalizes both the frameworks proposed by Bertsekas and Tsitsikli and by Bertsekas and Yu. We prove that the problem is well-defined and (weakly) polynomial when (i) there is a way to reach the target state from any initial state and (ii) there is no transition cycle of negative costs (a generalization of negative cost cycles). These assumptions generalize the standard assumptions for the deterministic shortest path problem and our framework encapsulates the latter problem (in contrast with prior works). In this new setting, we can show that (a) one can restrict to deterministic and stationary policies, (b) the problem is still (weakly) polynomial through linear programming, (c) Value Iteration and Policy Iteration converge, and (d) we can extend Dijkstra's algorithm

Efficient computation of exact solutions for quantitative model checking

Quantitative model checkers for Markov Decision Processes typically use finite-precision arithmetic. If all the coefficients in the process are rational numbers, then the model checking results are rational, and so they can be computed exactly. However, exact techniques are generally too expensive or limited in scalability. In this paper we propose a method for obtaining exact results starting from an approximated solution in finite-precision arithmetic. The input of the method is a description of a scheduler, which can be obtained by a model checker using finite precision. Given a scheduler, we show how to obtain a corresponding basis in a linear-programming problem, in such a way that the basis is optimal whenever the scheduler attains the worst-case probability. This correspondence is already known for discounted MDPs, we show how to apply it in the undiscounted case provided that some preprocessing is done. Using the correspondence, the linear-programming problem can be solved in exact arithmetic starting from the basis obtained. As a consequence, the method finds the worst-case probability even if the scheduler provided by the model checker was not optimal. In our experiments, the calculation of exact solutions from a candidate scheduler is significantly faster than the calculation using the simplex method under exact arithmetic starting from a default basis.Comment: In Proceedings QAPL 2012, arXiv:1207.055

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration