Solving disjunctive optimization problems by generalized semi-infinite optimization techniques

We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods, for our approach neither a conjunctive nor a disjunctive normal form is expected. Applying existing lower level reformulations for the corresponding semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally … Read more

A collision detection approach for maximizing the material utilization

We introduce a new method for a task of maximal material utilization, which is is to fit a flexible, scalable three-dimensional body into another aiming for maximal volume whereas position and shape may vary. The difficulty arises from the containment constraint which is not easy to handle numerically. We use a collision detection method to … Read more

On Calmness of the Argmin Mapping in Parametric Optimization Problems

Recently, Canovas et. al. (2013) presented an interesting result: the argmin mapping of a linear semi-infinite program under canonical perturbations is calm if and only if some associated linear semi-infinite inequality system is calm. Using classical tools from parametric optimization, we show that the if-direction of this condition holds in a much more general framework … Read more

Confidence Levels for CVaR Risk Measures and Minimax Limits

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

Primal-dual methods for solving infinite-dimensional games

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

Global Optimization of Generalized Semi-Infinite Programs via Restriction of the Right Hand Side

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

A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization

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

A branch and bound approach for convex semi-infinite programming

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

On the sufficiency of finite support duals in semi-infinite linear programming

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

Projection: A Unified Approach to Semi-Infinite Linear Programs and Duality in Convex Programming

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