In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a...

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an ε−dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the original doubly nonlinear equation as ε → 0. The ...

Optimization and inverse problems governed with partial differential equations are often formulated as constrained nonlinear programming problems via the Lagrange formalism. The nonlinearity is treated using the sequential quadratic programming. A numerical solution then hinges on an efficient iterative method for the resulting saddle–point systems. In this paper we apply a semi–monotonic augme...

We consider probabilistically constrained problems, in which the multivariate random variables are located in the right-hand sides. The objective function is linear, and its optimization is subject to a set of linear constraints as well as a joint probabilistic constraint enforcing that the joint fulfillment of a system of linear inequalities with random right-hand side variables be above a pre...

This paper presents a model for digital pheromone implementation of Particle Swarm Optimization (PSO) to solve constrained optimization problems. Digital pheromones are models simulating real pheromones produced by insects for communication to indicate a source of food or a nesting location. When integrated within PSO, this principle of communication and organization between swarm members offer...

Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method....

Augmented Lagrangian methods are effective tools for solving large-scale nonlinear programming problems. At each outer iteration a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When the penalty parameter becomes very large the subproblem is difficult, therefore the effectiveness ...

We present a new approach to a class of non-convex LMI-constrained problem in robust control theory. The problems we consider may be recast as the minimization of a linear objective subject to linear matrix inequality (LMI) constraints in tandem with non-convex constraints related to rank conditions. We solve these problems using an extension of the augmented Lagrangian technique. The Lagrangia...

We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the nonrelaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimizat...

In this paper we first provide a general formula of inclusion for the DiniHadamard ε-subdifferential of the difference of two functions and show that it becomes equality in case the functions are directionally approximately starshaped at a given point and a weak topological assumption is fulfilled. To this end we give a useful characterization of the Dini-Hadamard ε-subdifferential by means of ...