An Interior Point Method for Linear Programming, with n Active Set Flavor

It is now well established that, especially on large linearprogramming problems, the simplex method typically takes upa number of iterations considerably larger than recentinterior-points methods in order to reach a solution.On the other hand, at each iteration, the size of thelinear system of equations solved by the formercan be significantly less than that of the linearsystem solved by the latter.The algorithm proposed in this paper can be thought ofas a compromise between the two extremes: conceptuallyan interior-point method, it ignores, at each iteration,all constraints except those in a small "active set"(in the dual framework). For sake of simplicity, inthis first attempt, an affine scaling algorithm is usedand strong assumptions are made on the problem. Globaland local quadratic convergence is proved

Joint Scheduling and Routing for Ad-hoc Networks Under Channel State Uncertainty

We determine a joint link activation and routing policy that maximizes the stable throughput region of time-varying wireless ad-hoc networks with multiple commodities. In practice, the state of the channel process from the time it is observed till the time a transmission actually takes place can be significantly different. With this in mind, we introduce a stationary policy that takes scheduling and routing decisions based on a possibly inaccurate estimate of the true channel state. We show optimality of this policy within a broad class of link activation processes. In particular, processes in this class may be induced by any policy, possibly non-stationary, even anticipative and aware of the entire sample paths, including the future, of the arrival, estimated and true channel state processes, as long as it has no knowledge on the current true channel state, besides that available through the estimated channel state

CONSOLE: A CAD tandem for optimization-based design interacting with user-supplied simulators

CONSOLE employs a recently developed design methodology (International Journal of Control 43:1693-1721) which provides the designer with a congenial environment to express his problem as a multiple ojective constrained optimization problem and allows him to refine his characterization of optimality when a suboptimal design is approached. To this end, in CONSOLE, the designed formulates the design problem using a high-level language and performs design task and explores tradeoff through a few short and clearly defined commands. The range of problems that can be solved efficiently using a CAD tools depends very much on the ability of this tool to be interfaced with user-supplied simulators. For instance, when designing a control system one makes use of the characteristics of the plant, and therefore, a model of the plant under study has to be made available to the CAD tool. CONSOLE allows for an easy interfacing of almost any simulator the user has available. To date CONSOLE has already been used successfully in many applications, including the design of controllers for a flexible arm and for a robotic manipulator and the solution of a parameter selection problem for a neural network

A polynomial-time interior-point method for conic optimization, with inexact barrier evaluations

We consider a primal-dual short-step interior-point method for conic convex optimization problems for which exact evaluation of the gradient and Hessian of the primal and dual barrier functions is either impossible or prohibitively expensive. As our main contribution, we show that if approximate gradients and Hessians of the primal barrier function can be computed, and the relative errors in such quantities are not too large, then the method has polynomial worst-case iteration complexity. (In particular, polynomial iteration complexity ensues when the gradient and Hessian are evaluated exactly.) In addition, the algorithm requires no evaluation---or even approximate evaluation---of quantities related to the barrier function for the dual cone, even for problems in which the underlying cone is not self-dual

Universal Duality in Conic Convex Optimization

Given a primal-dual pair of linear programs, it is known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +infinity and -infinity. In contrast, for optimization problems over nonpolyhedral convex cones, a nonzero duality gap can exist even in the case where the primal and dual problems are both feasible. For a pair of dual conic convex programs, we provide simple conditions on the onstraint matricesand cone under which the duality gap is zero for every choice of linear objective function and ight-hand-side We refer to this property as niversal duality Our conditions possess the following properties: (i) they are necessary and sufficient, in the sense that if (and only if) they do not hold, the duality gap is nonzero for some linear objective function and ight-hand-side (ii) they are metrically and topologically generic; and (iii) they can be verified by solving a single conic convex program. As a side result, we also show that the feasible sets of a primal conic convex program and its dual cannot both be bounded, unless they are both empty, and we relate this to universal duality

A Simple Primal-Dual Feasible Interior-Point Method for Nonlinear Programming with Monotone Descent

We propose and analyze a primal-dual interior point method of the "feasible" type, with the additional property that the objective function decreases at each iteration. A distinctive feature of the method is the use of di#erent barrier parameter values for each constraint, with the purpose of better steering the constructed sequence away from non-KKT stationary points. Assets of the proposed scheme include relative simplicity of the algorithm and of the convergence analysis, strong global and local convergence properties, and good performance in preliminary tests. In addition, the initial point is allowed to lie on the boundary of the feasible set

Newton-KKT interior-point methods for indefinite quadratic programming

Two interior-point algorithms are proposed and analyzed, for the (local) Solution of (possibly) indefinite quadratic programming problems. They are of the Newton-KKT variety in that (Much like in the case of primal-dual algorithms for linear programming) search directions for the "primal" variables and the Karush-Kuhn-Tucker (KKT) multiplier estimates are components of the Newton (or quasi-Newton) direction for the Solution of the equalities in the first-order KKT conditions of optimality or a perturbed version of these conditions. Our algorithms are adapted from previously proposed algorithms for convex quadratic programming and general nonlinear programming. First, inspired by recent work by P. Tseng based on a "primal" affine-scaling algorithm (a la Dikin) [J. of Global Optimization, 30 (2004), no. 2, 285-300]. we consider a simple Newton-KKT affine-scaling algorithm. Then, a "barrier" version of the same algorithm is considered, which reduces to the affine-scaling version when the barrier parameter is set to zero at every iteration, rather than to the prescribed value. Global and local quadratic convergence are proved under nondegeneracy assumptions for both algorithms. Numerical results on randomly generated problems Suggest that the proposed algorithms may be of great practical interest

On the computation of the real Hurwitz-stability radius

Recently Qiu et al. obtained a computationally attractive formula for the evaluation of the real stability radius. This formula involves a global maximization over frequency. Here, for the Hurwitz stability case, we show that the frequency range can be limited to a certain finite interval. Numerical experimentation suggests that this interval is often reasonably small

Avoiding the Maratos Effect by Means of a Nomnonotone Line Search IL Inequality Constrained Problems - Feasible Iterates.

When solving inequality constrained optimization problems via Sequential Quadratic Programming (SQP), it is potentially advantageous to generate iterates that all satisfy the constraints: all quadratic programs encountered are then feasible and there is no need for a surrogate merit function. It has recently been shown that this is indeed possible, by means of a suitable perturbation of the original SQP iteration, without losing superlinear convergence. In this context, the well known Maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution. These difficulties have been accommodated by making use of a suitable modification of a bending" technique proposed by Mayne and Polak, requiring evaluation of the constraints function at an auxiliary point at each iteration. In part I of this two-part paper, it was shown that, when feasibility of the successive iterates is not required, the Maratos effect can be avoided by combining Mayne and Polak's technique with a nonmonotone line search proposed by Grippo, Lampariello and Lucidi in the context of unconstrained optimization, in such a way that, asymptotically, function evaluations are no longer performed at auxiliary points. In this second part, we show that feasibility can be restored without resorting to additional constraint evaluations, by adaptively estimating a bound on the second derivatives of the active constraints. Extension to constrained minimax problems is briefly discussed