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This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator. [ABSTRACT FROM AUTHOR]

The Fermat-Torricelli problem is an optimization problem associated with a finite subset $\{a_{j}\}_{j=1}^{q}$ of ℝ and a family $\{c_{j}\}_{j=1}^{q}$ of positive weights. The function F to be minimized is defined by $F(x)=\sum _{j=1}^{q}c_{j}\Vert x-a_{j}\Vert$. In this paper, we extend this problem to the case of volumes. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

The idea of containment sets (csets) is due to Walster and Hansen, and the theory is mainly due to the first author. Now that floating point computation with infinities is widely accepted, it is necessary to achieve the same for interval computation. The cset of a function over a set in its domain space is the set of all limits of normal function values over that set. Csets form a sound basis for defining a number of practical models for interval arithmetic that handle division by zero and related operations in an intuitive and consistent manner. Cset-based systems offer new opportunities for compiler optimization by rearranging interval expressions, achieving tighter bounds by reducing dependencies within the expression. This paper presents basic theory. It discusses division by zero, the case for a global flag to support ``loose'' evaluation, performance, and semantics. It presents numerical examples using a trial Matlab implementation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the smoothing of an implicit function defined by a nonsmooth underdetermined system of equations F(y, z) =0. We apply a class of parametrized smoothing methods to smooth F and investigate the limiting behavior of the implicit function solving the smoothed equations. In particular, we discuss the approximation of the Clarke generalized Jacobian of the implicit function when F is piecewise smooth. As an application, we present an analysis of the generalized Karush-Kuhn-Tucker conditions of different forms for a piecewise-smooth equality-constrained minimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general primal-dual nonlinear rescaling (PDNR) method for convex optimization with inequality constraints. We prove the global convergence of the PDNR method and estimate the error bounds for the primal and dual sequences. In particular, we prove that, under the standard second-order optimality conditions, the error bounds for the primal and dual sequences converge to zero with linear rate. Moreover, for any given ratio 0 < γ < 1, there is a fixed scaling parameter kγ > 0 such that each PDNR step shrinks the primal-dual error bound by at least a factor 0 < γ < 1, for any k ≥ kγ. The PDNR solver was tested on a variety of NLP problems including the constrained optimization problems (COPS) set. The results obtained show that the PDNR solver is numerically stable and produces results with high accuracy. Moreover, for most of the problems solved, the number of Newton steps is practically independent of the problem size. [ABSTRACT FROM AUTHOR]