We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in … Read more
We consider combinatorial optimization problems with uncertain objective functions. In the min-max-min robust optimization approach, a fixed number k of feasible solutions is computed such that the respective best of them is optimal in the worst case. The idea is to calculate a set of candidate solutions in a potentially expensive preprocessing and then select … Read more
The primal-dual hybrid gradient method (PDHG) originates from the Arrow-Hurwicz method, and it has been widely used to solve saddle point problems, particularly in image processing areas. With the introduction of a combination parameter, Chambolle and Pock proposed a generalized PDHG scheme with both theoretical and numerical advantages. It has been analyzed that except for … Read more
In this paper, we extend a recently proposed scenario decomposition algorithm (Ahmed (2013)) for risk-neutral 0-1 stochastic programs to the risk-averse setting. Specifically, we consider risk-averse 0-1 stochastic programs with objective functions based on coherent risk measures. Using a dual representation of a coherent risk measure, we first derive an equivalent minimax reformulation of the … Read more
Satisficing, as an approach to decision-making under uncertainty, aims at achieving solutions that satisfy the problem’s constraints as well as possible. Mathematical optimization problems that are related to this form of decision-making include the P-model of Charnes and Cooper (1963). In this paper, we propose a general framework of satisficing decision criteria, and show a … Read more
We consider multistage stochastic programming problems in which the random parameters have finite support, leading to optimization over a finite scenario set. We propose a hierarchy of bounds based on partitions of the scenario set into subsets of (nearly) equal cardinality. These expected partition (EP) bounds coincide with EGSO bounds provided by Sandikci et al. … Read more
This paper presents an algorithm that solves optimization problems on a matrix manifold $\mathcal{M} \subseteq \mathbb{R}^{m \times n}$ with an additional rank inequality constraint. The algorithm resorts to well-known Riemannian optimization schemes on fixed-rank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted … Read more
We discuss a general approach to building non-asymptotic confidence bounds for stochastic optimization problems. Our principal contribution is the observation that a Sample Average Approximation of a problem supplies upper and lower bounds for the optimal value of the problem which are essentially better than the quality of the corresponding optimal solutions. At the same … Read more
This paper is devoted to the study of unconstrained planar multiobjective location problems, where distances between points are defined by means of the Manhattan norm. By identifying all nonessential objectives, we develop an effective algorithm for generating the whole set of efficient solutions. We prove the correctness of this algorithm and present some computational results, … Read more
We first study the fast minimization properties of the trajectories of the second-order evolution equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \beta \nabla^2 \Phi (x(t))\dot{x} (t) + \nabla \Phi (x(t)) = 0, \end{equation*} where $\Phi : \mathcal H \to \mathbb R$ is a smooth convex function acting on a real Hilbert space $\mathcal H$, and … Read more