This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained … Read more
The trust region subproblem (the minimization of a quadratic objective subject to one quadratic constraint and denoted TRS) has many applications in diverse areas, e.g. function minimization, sequential quadratic programming, regularization, ridge regression, and discrete optimization. In particular, it determines the step in trust region algorithms for function minimization. Trust region algorithms are popular for … Read more
An algorithm for smooth nonlinear constrained optimization problems is described, in which a sequence of feasible iterates is generated by solving a trust-region sequential quadratic programming (SQP) subproblem at each iteration, and perturbing the resulting step to retain feasibility of each iterate. By retaining feasibility, the algorithm avoids several complications of other trust-region SQP approaches: … Read more
The goal of this paper is to organize some of the mathematical and algorithmic aspects of the recently proposed space-mapping technique for continuous optimization with expensive function evaluations. First, we consider the mapping from the fine space to the coarse space when the models are vector-valued functions and when the space-mapping (nonlinear) least-squares residual is … Read more
A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of … Read more
This paper presents the results of the application of a non-differentiable optimization method in connection with the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW is an extension of the Vehicle Routing Problem. In the VRPTW the service at each customer must start within an associated time window. The Shortest Path decomposition of the … Read more
A slack-based feasible interior point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust region methods require special attention. It is shown how the Cauchy point, which is often computed in trust region methods, must be modified so that the … Read more