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constraints

Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomial-time interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of ever-increasing size. Interior methods...

We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a self-concordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a non-heuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several non-heuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.

We propose a generic path-following scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic procedure in use, primal, dual or primal-dual. We show convergence in O( p n) iterations. We verify that the primal, dual and primal-dual algorithms satisfy the local quadratic convergence property. The method can be applied to solve the linear programming problem (with a feasible start) and to compute the analytic center of a bounded polytope. The generic path-following scheme easily extends to the logarithmic penalty barrier approach. Keywords Interior point method, method of centers, path-following, linear programming. This work has been completed with the support from the Swiss National Foundation for Scientific Research, grant 12-34002.92. 1 Introduction Shortly after Karmarkar's s...

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Traditional simplex-based algorithms for two-stage stochastic linear programscan be broadly divided into two classes: (a) those that explicitly exploit the structure of the equivalent large-scale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly exploit that structure. Algorithms of class (b) are in general preferred. In 1988, following the work of Karmarkar for general linear programs, Birge and Qi [10] proposed a specialization of Karmarkar's algorithm for two-stage stochastic linear programs. The algorithm of Birge and Qi [10] is the first interior point analog of class (a). Several other authors have studied related and different interior point analogs of class (a). Birge and Qi [10] also presented an analysis of the computational complexity of their algorithm. This analysis indicates that the computational complexity (in terms of total arithmetic operations) of their algorithm is in general smaller than that of the Karmarkar's ...

This paper presents a theory for constructing and computing velocity-adapted scale-space filters for spario-temporal image data.

We develop a long-step surface-following version of the method of analytic centers for the fractional-linear problem min {t0 | t0B(x) − A(x) ∈ H, B(x) ∈ K, x ∈ G}, where H is a closed convex domain, K is a convex cone contained in the recessive cone of H, G is a convex domain and B(·), A(·) are affine mappings. Tracing a twodimensional surface of analytic centers rather than the usual path of centers allows to skip the initial “centering ” phase of the path-following scheme. The proposed long-step policy of tracing the surface fits the best known overall polynomial-time complexity bounds for the method and, at the same time, seems to be more attractive computationally than the short-step policy, which was previously the only one giving good complexity bounds. 1

We present an interior-point method for convex fractional programming. The proposed algorithm converges in polynomial time, just as in the case of a convex problem, even though convex fractional programs are only pseudo-convex. More precisely, the rate of convergence is measured in terms of the area of two-dimensional convex sets C k containing the optimal points, and the area of C k is reduced by a constant factor c ! 1 at each iteration. The factor c depends only on the self-concordance parameter of a barrier function associated with the feasible set. We present an outline of a practical implementation of the proposed method, and we report results of a few numerical experiments. 1. Introduction Interior-point methods for the solution of nonlinear programming problems were already introduced in the 1950s and 1960s; see [6] and the references given there. In the 1970s, new and seemingly superior approaches, such as sequential quadratic programming techniques, were developed, and as a ...