## Exploiting Negative Curvature in Deterministic and Stochastic Optimization

This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although some prior work has established convergence results for algorithms that integrate both descent and negative curvature directions, there has not yet been numerical evidence showing that such methods offer significant performance improvements. In … Read more

## Algorithmic Differentiation for Piecewise Smooth Functions: A Case Study for Robust Optimization

This paper presents a minimization method for Lipschitz continuous, piecewise smooth objective functions based on algorithmic differentiation (AD). We assume that all nondifferentiabilities are caused by abs(), min(), and max(). The optimization method generates successively piecewise linearizations in abs-normal form and solves these local subproblems by exploiting the resulting kink structure. Both, the generation of … Read more

## An Augmented Lagrangian Proximal Alternating Method for Sparse Discrete Optimization Problems

In this paper, an augmented Lagrangian proximal alternating (ALPA) method is proposed for two class of large-scale sparse discrete constrained optimization problems in which a sequence of augmented Lagrangian subproblems are solved by utilizing proximal alternating linearized minimization framework and sparse projection techniques. Under the Mangasarian-Fromovitz and the basic constraint qualification, we show that any … Read more

## Foundations of gauge and perspective duality

Common numerical methods for constrained convex optimization are predicated on efficiently computing nearest points to the feasible region. The presence of a design matrix in the constraints yields feasible regions with more complex geometries. When the functional components are gauges, there is an equivalent optimization problem—the gauge dual– where the matrix appears only in the … Read more

## Exact Methods for Recursive Circle Packing

Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). … Read more

## Proximal Mapping for Symmetric Penalty and Sparsity

This paper studies a class of problems consisting of minimizing a continuously differentiable function penalized with the so-called $\ell_0$-norm over a symmetric set. These problems are hard to solve, yet prominent in many fields and applications. We first study the proximal mapping with respect to the $\ell_0$-norm over symmetric sets, and provide an efficient method … Read more

## Comparison of IP and CNF Models for Control of Automated Valet Parking Systems

In automated valet parking system, a central computer controls a number of robots which have the capability to move in two directions, under cars, lift a car up, carry it to another parking slot, and drop it. We study the theoretical throughput limitations of these systems: Given a car park layout, an initial configuration of … Read more

## A Branch and Bound Algorithm for Nonconvex Quadratic Optimization with Ball and Linear Constraints

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem … Read more

## Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy

A split feasibility formulation for the inverse problem of intensity-modulated radiation therapy (IMRT) treatment planning with dose-volume constraints (DVCs) included in the planning algorithm is presented. It involves a new type of sparsity constraint that enables the inclusion of a percentage-violation constraint in the model problem and its handling by continuous (as opposed to integer) … Read more

## Faster Estimation of High-Dimensional Vine Copulas with Automatic Differentiation

Vine copula is an important tool in modeling dependence structures of continuous-valued random variables. The maximum likelihood estimation (MLE) for vine copulas has long been considered computationally difficult in higher dimensions, even in 10 or 20 dimensions. Current computational practice, including the implementation in the state-of- the-art R package VineCopula, suffers from the bottleneck of … Read more