An interior-point and decomposition approach to multiple stage stochastic programming

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A robust primal-dual interior-point algorithm for nonlinear programs

10.1137/S1052623402400641SIAM Journal on Optimization1441163-118

Infeasible Full-Newton-Step Interior-Point Method for the Linear Complementarity Problems

In this tesis, we present a new Infeasible Interior-Point Method (IPM) for monotone Linear Complementarity Problem (LPC). The advantage of the method is that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. However, by suitable choice of parameters the iterates are forced to stay in the neighborhood of the central path, hence, still guaranteeing the global convergence of the method under strict feasibility assumption. The number of iterations necessary to find -approximate solution of the problem matches the best known iteration bounds for these types of methods. The preliminary implementation of the method and numerical results indicate robustness and practical validity of the method

Dual methods and approximation concepts in structural synthesis

Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins

Computing Optimal Experimental Designs via Interior Point Method

In this paper, we study optimal experimental design problems with a broad class of smooth convex optimality criteria, including the classical A-, D- and p th mean criterion. In particular, we propose an interior point (IP) method for them and establish its global convergence. Furthermore, by exploiting the structure of the Hessian matrix of the aforementioned optimality criteria, we derive an explicit formula for computing its rank. Using this result, we then show that the Newton direction arising in the IP method can be computed efficiently via Sherman-Morrison-Woodbury formula when the size of the moment matrix is small relative to the sample size. Finally, we compare our IP method with the widely used multiplicative algorithm introduced by Silvey et al. [29]. The computational results show that the IP method generally outperforms the multiplicative algorithm both in speed and solution quality

N\mathcal{N}IPM-HLSP: An Efficient Interior-Point Method for Hierarchical Least-Squares Programs

Hierarchical least-squares programs with linear constraints (HLSP) are a type of optimization problem very common in robotics. Each priority level contains an objective in least-squares form which is subject to the linear constraints of the higher priority hierarchy levels. Active-set methods (ASM) are a popular choice for solving them. However, they can perform poorly in terms of computational time if there are large changes of the active set. We therefore propose a computationally efficient primal-dual interior-point method (IPM) for HLSP's which is able to maintain constant numbers of solver iterations in these situations. We base our IPM on the null-space method which requires only a single decomposition per Newton iteration instead of two as it is the case for other IPM solvers. After a priority level has converged we compose a set of active constraints judging upon the dual and project lower priority levels into their null-space. We show that the IPM-HLSP can be expressed in least-squares form which avoids the formation of the quadratic Karush-Kuhn-Tucker (KKT) Hessian. Due to our choice of the null-space basis the IPM-HLSP is as fast as the state-of-the-art ASM-HLSP solver for equality only problems.Comment: 17 pages, 7 figure