Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo–Lampariello–Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.

A new active-set method for smooth box-constrained minimization is introduced. The algorithm combines an unconstrained method, including a new line-search which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm. Keywords: Box-constrained minimization, numerical methods, activeset strategies, Spectral Projected Gradient. 1

A practical algorithm for box-constrained optimization is introduced. The algorithm combines an active-set strategy with spectral projected gradient iterations. In the interior of each face a strategy that deals eciently with negative curvature is employed. Global convergence results are given. Numerical results are presented. Keywords: box constrained minimization, active set methods, spectral projected gradients, dogleg path methods. AMS Subject Classication: 49M07, 49M10, 65K, 90C06, 90C20. 1

A review is given of the underlying theory and recent developments in regard to the Barzilai-Borwein steepest descent method for large scale unconstrained optimization. One aim is to assess why the method seems to be comparable in practical eciency to conjugate gradient methods. The importance of using a non-monotone line search is stressed, although some suggestions are made as to why the modi- cation proposed by Raydan [22] often does not usually perform well for an illconditioned problem. Extensions for box constraints are discussed. A number of interesting open questions are put forward. Keywords Barzilai-Borwein method, steepest descent, elliptic systems, unconstrained optimization. 1

Abstract. An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong secondorder sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries. Key words. nonmonotone gradient projection, box constrained optimization, active set algorithm,

A new primal-dual algorithm is proposed for the minimization of non-convex objective functions subject to simple bounds and linear equality constraints. The method alternates between a classical primal-dual step and a Newton-like step in order to ensure descent on a suitable merit function. Convergence of a well-defined subsequence of iterates is proved from arbitrary starting points. Algorithmic variants are discussed and preliminary numerical results presented. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA Email : arconn@watson.ibm.com 2 Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU Email : nimg@letterbox.rl.ac.uk 3 Current reports available by anonymous ftp from joyous-gard.cc.rl.ac.uk (internet 130.246.9.91) in the directory "pub/reports". 4 Department of Mathematics, Facult'es Universitaires ND de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium, EU Email : pht@ma...

Noise can cause digital circuits to switch incorrectly and thus produce spurious results. Noise can also have adverse power, timing and reliability e ects. Dynamic logic is particularly susceptible to charge-sharing and coupling noise. Thus the design and optimization of a circuit should take noise considerations into account. Such considerations are typically stated as semi-in nite constraints. In addition, the number of signals to be checked and the number of sub-intervals of time during which the checking must be performed can potentially be very large. Thus, the practical incorporation of noise constraints during circuit optimization is a hitherto unsolved problem. This paper describes a novel method for incorporating noise considerations during automatic circuit optimization. Semi-in nite constraints representing noise considerations are rst converted toordinary equality constraints involving time integrals, which are readily computed in the context of circuit optimization based on time-domain simulation. Next, the gradients of these integrals are computed by the adjoint method. By using an augmented Lagrangian optimization merit function, the adjoint method is applied tocompute all the necessary gradients required for optimization in a single adjoint analysis, no matter how many noise measurements are considered and irrespective of the dimensionality of the problem. Numerical results are presented. 1

Global convergence to first-order critical points is proved for two trust-region SQP-filter algorithms of the type introduced by Fletcher and Leyffer (1997). The algorithms allow for an approximate solution of the quadratic subproblem and incorporate the safeguarding tests described in Fletcher, Leyffer and Toint (1998). The first algorithm decomposes the step into its normal and tangential components, while the second replaces this decomposition by a stronger condition on the associated model decrease. 1 Department of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, EU. Email : fletcher@mcs.dundee.ac.uk, sleyffer@mcs.dundee.ac.uk 2 Current reports available from "http://www.mcs.dundee.ac.uk:8080/~dfg/Narep.html". 3 Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU. Email : n.gould@rl.ac.uk 4 Current reports available from "http://www.numerical.rl.ac.uk/reports/reports.html". 5 Department ...

this paper is to show the application of these canonical formulas to optimal control processes being integrated by the Runge-Kutta family of numerical methods. There are many papers concerning numerical comparisions between automatic differentiation, finite differences and symbolic differentiation. See, for example, [1, 2, 6, 7, 21] among others. Another objective is to test the behavior of the spectral projected gradient methods introduced in [5]. These methods combine the classical projected gradient with two recently developed ingredients in optimization: (i) the nonmonotone line search schemes of Grippo, Lampariello and Lucidi ([24]), and (ii) the spectral steplength (introduced by Barzilai and Borwein ([3]) and analyzed by Raydan ([30, 31])). This choice of the steplength requires little computational work and greatly speeds up the convergence of gradient methods. The numerical experiments presented in [5], showing the high performance of these fast and easily implementable methods, motivate us to combine the spectral projected gradient methods with automatic differentiation. Both tools are used in this work for the development of codes for numerical solution of optimal control problems. In Section 2 of this paper, we apply the canonical formulas to the discrete version of the optimal control problem. In Section 3, we give a concise survey about spectral projected gradient algorithms. Section 4 presents some numerical experiments. Some final remarks are presented in Section 5. 2 CANONICAL FORMULAS The basic optimal control problem can be described as follows: Let a process governed by a system of ordinary differential equations be dx(t) dt = f(x(t); u(t); ); T 0 t T f ; (1) where x : [T 0 ; T f ] ! IR nx , u : [T 0 ; T f ] ! U ` IR nu , U compact, and 2 V ...

. We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture. Key words. Horizontal linear complementarity problem, linear complementarity problem, bound constrained minimization, optimality conditions, stationary points, global minimizers. AMS (MOS) subject classification. 49M15, 65K05, 90C33. Published in Journal of Global Optimization 6, 1995, pp. 1-15. This work was supported by FAPESP (grants 9...