Conditional value at risk (CVaR) has been widely used as a risk measure in finance. When the confidence level of CVaR is set close to 1, the CVaR risk measure approximates the extreme (worst scenario) risk measure. In this paper, we present a quantitative analysis of the relationship between the two risk measures and its … Read more
Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre’s semidefinite programming hierarchy. We generalize … Read more
We consider the natural generalizations of blocking and anti-blocking polyhedra in infinite dimensions, and study issues related to duality and integrality of extreme points for these sets. Using appropriate finite truncations, we give conditions under which complementary slackness holds for primal-dual pairs of the infi nite linear programming problems associated with infi nite blocking and anti-blocking polyhedra. … Read more
In this paper we show that the infinite-dimensional differential games with simple objective functional can be solved in a finite-dimensional dual form in the space of dual multipliers for the constraints related to the end points of the trajectories. The primal solutions can be easily reconstructed by the appropriate dual subgradient schemes. The suggested schemes … Read more
The algorithm proposed in [Mitsos Optimization 2011] for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs (GSIP). No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) … Read more
We first present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems, and use it to develop an algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. The cutting surface algorithm is also applicable to problems with non-differentiable semi-infinite … Read more
In this paper we propose an efficient approach for globally solving a class of convex semi-infinite programming (SIP) problems. Under the objective function and constraints (w.r.t. the variables to be optimized) convexity assumption, and appropriate differentiability, we propose a branch and bound exchange type method for SIP. To compute a feasible point for a SIP … Read more
We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash~\cite{anderson-nash}. … Read more
Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. We extend Fourier-Motzkin elimination to semi-infinite linear programs which are linear programs with finitely many variables and infinitely many constraints. Applying projection leads to new characterizations of important properties for primal-dual pairs of semi-infinite programs such as zero duality gap, feasibility, boundedness, and solvability. … Read more
Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide … Read more