Bilevel programs form a class of hierarchical optimzation problems in which the constraint set is not explicit but is defined in terms of another optimization problem. Optimization problems in several real-life domains can be expressed as bilevel programs, making such problems ubiquitous. In this term paper, we analyze a trust region approach for solving a special case of this problem.

Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for lower level problem resulting in a mathematical program with complementarity constraints (MPCC). First, we present KKT reformulation of bilevel optimization on Riemannian manifolds. Moreover, show that global optimal solutions theMPCCcorrespond to manifolds provided convex satisfies Slater?s constrai...

Support vector machines and related classification models require the solution of convex optimization problems that have one or more regularization hyper-parameters. Typically, the hyper-parameters are selected to minimize cross validated estimates of the out-of-sample classification error of the model. This cross-validation optimization problem can be formulated as a bilevel program in which t...

We investigate a special mixed-discrete bilevel programming problem resulting from the task to compute cash-out penalties due to the supply of incorrect amounts of natural gas through large gas networks. For easier solution we move the discreteness demand from the lower to the upper levels. This clearly changes the problem and it is our aim to investigate the relations between both problems. Af...

The present paper serves to introduce a new robust modelling technique and solution methodology for a class of problems in structural optimization, in which we calculate the structural design that responds the best on average to a given set of loads, each having its own probability of occurrence. A very general stochastic model is presented, along with its mathematical properties, generalizing ...

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of gener...

The bilevel programming (BP) problem is a hierarchical optimization problem where a subset of the variables is constrained to be a solution of a given optimization problem pa-rameterized by the remaining variables. The BP problem is a multilevel programming problem with two levels. The hierarchical optimization sctructure appears naturally in many applications when lower level actions depend on...

This paper presents a voltage stability constrained optimal power flow that is expressed via a bilevel programming framework. The inner objective function is dedicated for maximizing voltage stability margin while the outer objective function is focused on minimization of total production cost of thermal units. The original two stage problem is converted to a single level optimization problem v...

ABSTRACT: Feasible solutions of Bilevel programming (BLP) problem have to satisfy the optimality conditions of a lower level optimization problem which is defined as part of its constraints. Multiobjective BLP (MOBLP) problems are the special case of BLP problems with multiple objectives at the upper level which require every feasible solution to satisfy the Pareto optimality conditions of a lo...