For controlled discrete-time stochastic processes we introduce a new class of dynamic risk measures, which we call process-based. Their main features are that they measure risk of processes that are functions of the history of the base process. We introduce a new concept of conditional stochastic time consistency and we derive the structure of process-based … Read more
We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a given collection of subsets (regions or neighborhoods) in the underlying metric space. We give a randomized polynomial time approximation scheme (PTAS) when the regions are fat weakly disjoint. This notion … Read more
Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., problems of the form $\min_x \E_v\[f_v\big(\E_w [g_w(x)]\big) \]$. In order to solve this stochastic composition problem, we propose a class … Read more
In this paper two simple examples of a twice continuously differentiable strictly convex function $f$ are presented for which Newton’s method with line search converges to a point where the gradient of $f$ is not zero. The first example uses a line search based on the Wolfe conditions. For the second example, some strictly convex … Read more
The rank minimization with linear equality constraints has two closely related models, the low rank approximation model, that is to find the best rank-k approximation of a matrix satisfying the linear constraints, and its corresponding factorization model. The latter one is an unconstrained nonlinear least squares problem and hence enjoys a few fast first-order methods … Read more
As an essential substructure underlying a large class of chance-constrained programming problems with finite discrete distributions, the mixing set with $0-1$ knapsack has received considerable attentions in recent literature. In this study, we present a family of strong inequalities that subsume existing ones for this set. We also find many other inequalities that can be … Read more
We propose a limited memory steepest descent (LMSD) method for solving unconstrained optimization problems. As a steepest descent method, the step computation in each iteration requires the evaluation of a gradient of the objective function and the calculation of a scalar step size only. When employed to solve certain convex problems, our method reduces to … Read more
Designing algorithms for an optimization model often amounts to maintaining a balance between the degree of information to request from the model on the one hand, and the computational speed to expect on the other hand. Naturally, the more information is available, the faster one can expect the algorithm to converge. The popular algorithm of … Read more
In this article we establish new second order necessary and sufficient optimality conditions for a class of control-affine problems with a scalar control and a scalar state constraint. These optimality conditions extend to the constrained state framework the Goh transform, which is the classical tool for obtaining an extension of the Legendre condition. We propose … Read more
We address the inverse problem of Lagrangian identification based on trajectories in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advantages from the theoretical, numerical and statistical points of … Read more