In this paper, we propose a filter active-set algorithm for the minimization problem over a product of multiple ball/sphere constraints. By making effective use of the special structure of the ball/sphere constraints, a new limited memory BFGS (L-BFGS) scheme is presented. The new L-BFGS implementation takes advantage of the sparse structure of the Jacobian of … Read more
In this paper we study proximal conditional-gradient (CG) and proximal gradient-projection type algorithms for a block-structured constrained nonconvex optimization model, which arises naturally from tensor data analysis. First, we introduce a new notion of $\epsilon$-stationarity, which is suitable for the structured problem under consideration. %, compared with other similar solution concepts. We then propose two … Read more
Due to strict regulatory rules in combination with complex nonlinear physics, major gas network operators in Germany and Europe face hard planning problems that call for optimization. In part 1 of this paper we have developed a suitable model hierarchy for that purpose. Here we consider the more practical aspects of modeling. We validate individual … Read more
A class of trust-region algorithms is developed and analyzed for the solution of optimization problems with nonlinear equality and inequality constraints. Based on composite-step trust region methods and a filter approach, the resulting algorithm also does not require the computation of exact Jacobians; only Jacobian vector products are used along with approximate Jacobian matrices. As … Read more
We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed … Read more
We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to an objective function value) to one returned … Read more
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
This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more
Quadratic models Q_k(.) of the objective function F(.) are used by many successful iterative algorithms for minimization, where k is the iteration number. Given the vector of variables x_k, a new vector x_{k+1} may be calculated that satisfies Q_k(x_{k+1}) < Q_k(x_k), in the hope that it provides the reduction F(x_{k+1}) < F(x_k). Trust region methods ... Read more
In this work we present a new feasible direction algorithm for solving smooth nonlinear second-order cone programs. These problems consist of minimizing a nonlinear di erentiable objective function subject to some nonlinear second-order cone constraints. Given a point interior to the feasible set de nfined by the nonlinear constraints, the proposed approach computes a feasible and descent … Read more