Primal Feasible Point Research Articles

Safe Feature Elimination for Non-negativity Constrained Convex Optimization

Inspired by recent work on safe feature elimination for $1$-norm regularized least-squares, we develop strategies to eliminate features from convex optimization problems with non-negativity constraints. Our strategy is safe in the sense that it will only remove features/coordinates from the problem when they are guaranteed to be zero at a solution. To perform feature elimination we use an accurate, but not optimal, primal-dual feasible pair, making our methods robust and able to be used on ill-conditioned problems. We supplement our feature elimination problem with a method to construct an accurate dual feasible point from an accurate primal feasible point; this allows us to use a first-order method to find an accurate primal feasible point, then use that point to construct an accurate dual feasible point and perform feature elimination. Under reasonable conditions, our feature elimination strategy will eventually eliminate all zero features from the problem. As an application of our methods we show how safe feature elimination can be used to robustly certify the uniqueness of non-negative least-squares (NNLS) problems. We give numerical examples on a well-conditioned synthetic NNLS problem and a on set of 40000 extremely ill-conditioned NNLS problems arising in a microscopy application.

A Convex Approximation for Optimal DER Scheduling on Unbal-anced Power Distribution Networks

The increase of solar photovoltaic penetration poses several challenges for distribution network operation, mainly because such high penetration might cause reliability problems like protection malfunctioning, accelerated decay of voltage regulators and voltage violations. Some control strategies avoid such drawbacks at the cost of not fully exploiting the available energy. Existing solutions based on mathematical programming solve a 3-phase ACOPF to optimally exploit the available energy, however, this might increase all reliability problems above if done carelessly. As a solution to optimally exploit DERs (like local photovoltaic and storage systems) without compromising the network reliability, this paper presents a novel algorithm to solve the 3-phase ACOPF as a sequence of convex Quadratically Constrained Quadratic Programs. Results show that this solution has a lower voltage unbalance and computation time than its non-linear counterpart, furthermore, it converges to a primal feasible point for the non-linear formulation without major sacrifices on optimal DER active power injections.

On the Convergence of Alternating Direction Lagrangian Methods for Nonconvex Structured Optimization Problems

Nonconvex and structured optimization problems arise in many engineering applications that demand scalable and distributed solution methods. The study of the convergence properties of these methods is, in general, difficult due to the nonconvexity of the problem. In this paper, two distributed solution methods that combine the fast convergence properties of augmented Lagrangian-based methods with the separability properties of alternating optimization are investigated. The first method is adapted from the classic quadratic penalty function method and is called the alternating direction penalty method (ADPM). Unlike the original quadratic penalty function method, where single-step optimizations are adopted, ADPM uses an alternating optimization which, in turn, makes it scalable. The second method is the well-known alternating direction method of multipliers (ADMM). It is shown that ADPM for nonconvex problems asymptotically converges to a primal feasible point under mild conditions and an additional condition ensuring that it asymptotically reaches the standard first-order necessary conditions for local optimality is introduced. In the case of the ADMM, novel sufficient conditions under which the algorithm asymptotically reaches the standard first-order necessary conditions are established. Based on this, complete convergence of the ADMM for a class of low-dimensional problems is characterized. Finally, the results are illustrated by applying ADPM and ADMM to a nonconvex localization problem in wireless-sensor networks.

A deficient-basis dual counterpart of Paparrizos, Samaras and Stephanides’ primal-dual simplex-type algorithm†

In a recent primal-dual simplex-type algorithm (K. Paparrizos, N. Samaras and G. Stephanides, A new efficient primal dual simplex algorithm, Computers & Operations Research 30 (2003), pp. 1383–1399), its authors show how to take advantage of the knowledge of a primal feasible point and they work with a square basis during the whole process. In this paper we address what could be thought of as its deficient-basis dual counterpart by showing how to take advantage of the knowledge of a dual feasible point in a deficient-basis simplex-type environment. Three small examples are given to illustrate how the specific pivoting rules designed for the proposed algorithm deal with non-unique dual solutions, unbounded dual objectives and a classical exponential example by Goldfarb, thus avoiding some caveats of the dual simplex method. Practical experiments with a new collection of difficult problems for the dual simplex method are reported to justify iteration decrease, and we sketch some details of a sparse projected-gradient implementation in terms of Davis and Hager's CHOLMOD sparse Cholesky factorization (which is row updatable/downdatable) to solve the underlying least-squares subproblems, namely linear least-squares problems and projections onto linear manifolds. †Review (November 2008) of Technical Report MA-07/01 (http://www.matap.uma.es/investigacion/tr.html), Department Applied Mathematics, University of Málaga, September 2007. Talk presented at Joint 2nd Conference on Optimization Methods & Software and 6th EUROPT Workshop on Advances in Continuous Optimization, Prague, Czech Republic, July 2007.

A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example.

A central cutting plane algorithm for the convex programming problem

An algorithm is developed for solving the convex programming problem which iteratively proceeds to the optimum by constructing a cutting plane through the center of a polyhedral approximation to the optimum. This generates a sequence of primal feasible points whose limit points satisfy the Kuhn—Tucker conditions of the problem. Additionally, we present a simple, effective rule for dropping prior cuts, an easily calculated bound on the objective function, and a rate of convergence.

Technical Note—Computational Experience with Normed and Nonnormed Column-Generation Procedures in Nonlinear Programming

Nemhauser and Widhelm have suggested a normed variant of the Dantzig-Wolfe convex programming algorithm. The normalization explicitly occurs in the multiplier-space. There the interior point used to generate the next cut is one that maximizes the minimum slack from the present cuts to that point. Nemhauser and Widhelm suggested this normalization for two reasons. First, since the corresponding cut at worst includes the generating point, choosing that point “centered” in some geometric sense should result in a generally efficient reduction in the feasible search region. Second, cuts not active in defining the present feasibility region can affect the choice of the interior-point if norming is not applied. This note demonstrates how primal feasible points can be generated at each iteration and cites limited computational results indicating that the normed procedure is generally more efficient than the original algorithm from the standpoint of both the number of iterations and computational time.