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In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied. [ABSTRACT FROM AUTHOR]

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate descent approach, together with variants and extensions and their convergence properties, mostly with reference to convex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient implementations of accelerated coordinate descent algorithms are possible for problems of this type. We also present some parallel variants and discuss their convergence properties under several models of parallel execution. [ABSTRACT FROM AUTHOR]

In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the $$k$$ -robust model where the possible scenarios tomorrow are given by all demand-subsets of size $$k$$ . In this paper, we give the following simple and intuitive template for $$k$$ -robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for $$k$$ -robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our $$k$$ -robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, which are based on dual-rounding and primal-dual ideas; these properties might be of some independent interest. As a by-product of our techniques, we also get algorithms for max-min problems of the form: ' given a covering problem instance, which $$k$$ of the elements are costliest to cover?' For the problems mentioned above, we show that their $$k$$ -max-min versions have performance guarantees similar to those for the $$k$$ -robust problems. [ABSTRACT FROM AUTHOR]

This paper presents an active-set algorithm for large-scale optimization that occupies the middle ground between sequential quadratic programming and sequential linear-quadratic programming methods. It consists of two phases. The algorithm first minimizes a piecewise linear approximation of the Lagrangian, subject to a linearization of the constraints, to determine a working set. Then, an equality constrained subproblem based on this working set and using second derivative information is solved in order to promote fast convergence. A study of the local and global convergence properties of the algorithm highlights the importance of the placement of the interpolation points that determine the piecewise linear model of the Lagrangian. [ABSTRACT FROM AUTHOR]

Optimality functions define stationarity in nonlinear programming, semi-infinite optimization, and optimal control in some sense. In this paper, we consider optimality functions for stochastic programs with nonlinear, possibly nonconvex, expected value objective and constraint functions. We show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer. We construct confidence intervals for the value of the optimality function at a candidate point and, hence, provide a quantitative measure of solution quality. Based on sample average approximations, we develop an algorithm for classes of stochastic programs that include CVaR-problems and utilize optimality functions to select sample sizes. [ABSTRACT FROM AUTHOR]

We perform a smoothed analysis of the GCC-condition number $${\fancyscript{C}(A)}$$ of the linear programming feasibility problem $${\exists x\in\mathbb{R}^{m+1}\,Ax < 0}$$. Suppose that $${\bar{A}}$$ is any matrix with rows $${\overline{a}_i}$$ of euclidean norm 1 and, independently for all i, let a be a random perturbation of $${\overline{a}_i}$$ following the uniform distribution in the spherical disk in S of angular radius arcsin σ and centered at $${\overline{a}_i}$$. We prove that $${\mathbb{E}({\rm ln}\,\fancyscript{C}(A)) =\mathcal{O} (mn/\sigma)}$$. A similar result was shown for Renegar's condition number and Gaussian perturbations by Dunagan et al. (Math Program Ser A, 2009). Our result is robust in the sense that it easily extends to radially symmetric probability distributions supported on a spherical disk of radius arcsin σ, whose density may even have a singularity at the center of the perturbation. Our proofs combine ideas from a recent paper of Bürgisser et al. (Math Comp 77(263), 2008) with techniques of Dunagan et al. [ABSTRACT FROM AUTHOR]

Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization. [ABSTRACT FROM AUTHOR]