Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available, and that the constraint gradients are sparse.

This paper is an invited contribution to the 50th anniversary issue of the journal Operations Research, published by the Institute of Operations Research and Management Science (INFORMS). It describes one persons perspective on the development of computational tools for linear programming. The paper begins with a short, personal history, followed by historical remarks covering the some 40 years of linear-programming developments that predate my own involvement in this subject. It concludes with a more detailed look at the evolution of computational linear programming since 1987. 2

This paper presents a nonparametric penalized likelihood approach for variable selection and model building, called likelihood basis pursuit (LBP). In the setting of a tensor product reproducing kernel Hilbert space, we decompose the log likelihood into the sum of different functional components such as main effects and interactions, with each component represented by appropriate basis functions. The basis functions are chosen to be compatible with variable selection and model building in the context of a smoothing spline ANOVA model. Basis pursuit is applied to obtain the optimal decomposition in terms of having the smallest l 1 norm on the coefficients. We use the functional L 1 norm to measure the importance of each component and determine the "threshold" value by a sequential Monte Carlo bootstrap test algorithm. As a generalized LASSO-type method, LBP produces shrinkage estimates for the coefficients, which greatly facilitates the variable selection process, and provides highly interpretable multivariate functional estimates at the same time. To choose the regularization parameters appearing in the LBP models, generalized approximate cross validation (GACV) is derived as a tuning criterion. To make GACV widely applicable to large data sets, its randomized version is proposed as well. A technique "slice modeling" is used to solve the optimization problem and makes the computation more efficient. LBP has great potential for a wide range of research and application areas such as medical studies, and in this paper we apply it to two large on-going epidemiological studies: the Wisconsin Epidemiological Study of Diabetic Retinopathy (WESDR) and the Beaver Dam Eye Study (BDES).

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Abstract. We discuss an implementation of a derivative-free generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints. The convergence guarantees for generating set search methods require that the set of search directions possesses certain geometrical properties that allow it to approximate the feasible region near the current iterate. In the hard case, the calculation of the search directions corresponds to finding the extreme rays of a cone with a degenerate vertex at the origin, a difficult problem. We discuss here how state-of-the-art computational geometry methods make it tractable to solve this problem in connection with generating set search. We also discuss a number of other practical issues of implementation, such as the careful treatment of equality constraints and the desirability of augmenting the set of search directions beyond the theoretically minimal set. We illustrate the behavior of the implementation on several problems from the CUTEr test suite. We have found it to be successful on problems with several hundred variables and linear constraints.

Abstract — Updating probabilistic belief matrices as new observations arrive, in the presence of noise, is a critical part of many algorithms for target tracking in sensor networks. These updates have to be carried out while preserving sum constraints, arising for example, from probabilities. This paper addresses the problem of updating belief matrices to satisfy sum constraints using scaling algorithms. We show that the convergence behavior of the Sinkhorn scaling process, used for scaling belief matrices, can vary dramatically depending on whether the prior unscaled matrix is exactly scalable or only almost scalable. We give an efficient polynomial-time algorithm based on the maximum-flow algorithm that determines whether a given matrix is exactly scalable, thus determining the convergence properties of the Sinkhorn scaling process. We prove that the Sinkhorn scaling process always provides a solution to the problem of minimizing the Kullback-Leibler distance of the physically feasible scaled matrix from the prior constraint-violating matrix, even when the matrices are not exactly scalable. We pose the scaling process as a linearly constrained convex optimization problem, and solve it using an interior-point method. We prove that even in cases in which the matrices are not exactly scalable, the problem can be solved to ɛ−optimality in strongly polynomial time, improving the best known bound for the problem of scaling arbitrary nonnegative rectangular matrices to prescribed row and column sums. I.

An interior-point algorithm within a parallel branch-and-bound framework for solving nonlinear mixed integer programs is described. The nonlinear programming relaxations at each node are solved using an interior point SQP method. In contrast to solving the relaxation to optimality at each tree node, the relaxation is only solved to near-optimality. Analogous to employing advanced bases in simplex-based linear MIP solvers, a “dynamic” collection of warmstart vectors is kept to provide “advanced warmstarts” at each branch-and-bound node. The code has the capability to run in both shared-memory and distributed-memory parallel environments. Preliminary computational results on various classes of linear mixed integer programs and quadratic portfolio problems are presented.

We examine the influence of optimality measures on the benchmarking process, and show that scaling requirements lead to a convergence test for nonlinearly constrained solvers that uses a mixture of absolute and relative error measures. We show that this convergence test is well behaved at any point where the constraints satisfy the Mangasarian-Fromovitz constraint qualification and also avoids the explicit use of a complementarity measure. Our computational experiments explore the impact of this convergence test on the benchmarking process with performance profiles. 1

Networks, such as the electric grid, are operated by sets of agents that are heterogeneous, local and distributed. (By "heterogeneous" we mean that the agents can range from simple devices, like relays, to very intelligent entities, like committees of humans. By "local and distributed" we mean that each agent can sense only a few of the network's state variables and influence only a few of its control variables.) We are concerned with two issues: the quality and speed of decision-making by heterogeneous, local and distributed agents. For quality, our standard of comparison is an ideal, centralized agent, which senses the state of the entire network and makes globally optimal decisions. (Of course, such a centralized agent is impractical for large networks.