From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

Successive Optimization Method via Parametric Monotone Composition Formulation*

In this paper a successive optimization method for solving inequality constrained optimization problems is introduced via a parametric monotone composition reformulation. The global optimal value of the original constrained optimization problem is shown to be the least root of the optimal value function of an auxiliary parametric optimization problem, thus can be found via a bisection method. The parametric optimization subproblem is formulated in such a way that it is a one-parameter problem and its value function is a monotone composition function with respect to the original objective function and the constraints. Various forms can be taken in the parametric optimization problem in accordance with a special structure of the original optimization problem, and in some cases, the parametric optimization problems are convex composite ones. Finally, the parametric monotone composite reformulation is applied to study local optimality.     

A convexification method for a class of global optimization problems with applications to reliability optimization

A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.     

A global optimization algorithm for reliable network design

In this paper, we consider the problem of designing reliable networks that satisfy supply/demand, flow balance, and capacity constraints, while simultaneously allocating certain resources to mitigate the arc failure probabilities in such a manner as to minimize the total cost of network design and resource allocation. The resulting model formulation is a nonconvex mixed-integer 0-1 program, for which a tight linear programming relaxation is derived using RLT-based variable substitution strategies and a polyhedral outer-approximation technique. This LP relaxation is subsequently embedded within a specialized branch-and-bound procedure, and the proposed approach is proven to converge to a global optimum. Various alternative partitioning strategies that could potentially be employed in the context of this branch-and-bound framework, while preserving the theoretical convergence property, are also explored. Computational results are reported for a hypothetical scenario based on different parameter inputs and alternative branching strategies. Related optimization models that conform to the same class of problems are also briefly presented.