this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can

Abstract. Large scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. The state-of-the-art for such problems is reduced quasi-Newton sequential quadratic programming (SQP) methods. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this two-part article we propose a new method for steady-state PDE-constrained optimization, based on the idea of full space SQP with reduced space quasi-Newton SQP preconditioning. The basic components of the method are: Newton solution of the first-order optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the Karush-Kuhn-Tucker (KKT) linear systems arising at each Newton iteration using a symmetric quasi-minimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation or by a two-step stationary method. Accordingly, we term the new method Lagrange-Newton-Krylov Schur (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In the first part of the paper we investigate the effectiveness of the KKT linear system solver. We test the method on two optimal control problems in which the flow is described by the steady-state Stokes equations. The

In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [34] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numerical tests.

Successful treatment of inconsistent QP problems is of major importance in the SQP method, since such occur quite often even for well behaved nonlinear programming problems. This paper presents a new technique for regularizing inconsistent QP problems, which compromises in its properties between the simple technique of Pantoja and Mayne [34] and the highly successful, but expensive one of Tone [44]. Global convergence of a corresponding algorithm is shown under reasonable weak conditions. Numerical results are reported which show that this technique, combined with a special method for the case of regular subproblems, is quite competitive to highly appreciated established ones. Key words: sequential quadratic programming, SQP method, nonlinear programming AMS(MOS) subject classification: primary 90C30, secondary 65K05 1 NOTATION Superscripts on a vector denote elements of sequences. All vectors are column vectors. For a vectorvalued function g rg(x) denotes the transposed Jacobian eval...

The need for improvement in process operations, logistics and supply chain management has created a great demand for the development of optimization models for planning and scheduling. In this paper we first review the major classes of planning and scheduling models that arise in process operations, and establish the underlying mathematical structure of these problems. As will be shown, the nature of these models is greatly affected by the time representation (discrete or continuous), and is often dominated by discrete decisions. We then briefly review the major recent developments in mixed-integer linear and nonlinear programming, disjunctive programming and constraint programming, as well as general decomposition techniques for solving these problems. We present a general formulation for integrating planning and scheduling to illustrate the models and methods discussed in this paper.

An algorithm for general nonlinearly constrained optimization is presented, which solves an unconstrained piecewise quadratic subproblem and a quadratic programming subproblem at each iterate. The algorithm is robust since it can circumvent the difficulties associated with the possible inconsistency of QP subproblem of the original SQP method. Moreover, the algorithm can converge to a point which satisfies a certain first-order necessary optimality condition even when the original problem is itself infeasible, which is a feature of Burke and Han's methods(1989). Unlike Burke and Han's methods(1989), however, we do not introduce additional bound constraints. The algorithm solves the same subproblems as Han-Powell SQP algorithm at feasible points of the original problem. Under certain assumptions, it is shown that the algorithm coincide with the Han-Powell method when the iterates are sufficiently close to the solution. Some global convergence results are proved and local superlinear co...

Standard global convergence proofs are examined to determine why some algorithms perform better than other algorithms. We show that relaxing the conditions required to prove global convergence can improve an algorithm's performance. Further analysis indicates that minimizing an estimate of the distance to the minimum relaxes the convergence conditions in such a way as to improve an algorithm's convergence rate. A new line-search algorithm based on these ideas is presented that does not force a reduction in the objective function at each iteration, yet it allows the objective function to increase during an iteration only if this will result in faster convergence. Unlike the nonmonotone algorithms in the literature, these new functions dynamically adjust to account for changes between the influence of curvature and descent. The result is an optimal algorithm in the sense that an estimate of the distance to the minimum is minimized at each iteration. The algorithm is shown to be well defi...

. Exact penalty methods for the solution of constrained optimization problems are based on the construction of a function whose unconstrained minimizing points are also solution of the constrained problem. In the first part of this paper we recall some definitions concerning exactness properties of penalty functions, of barrier functions, of augmented Lagrangian functions, and discuss under which assumptions on the constrained problem these properties can be ensured. In the second part of the paper we consider algorithmic aspects of exact penalty methods; in particular we show that, by making use of continuously differentiable functions that possess exactness properties, it is possible to define implementable algorithms that are globally convergent with superlinear convergence rate towards KKT points of the constrained problem. 1 Introduction "It would be a major theoretic breakthrough in nonlinear programming if a simple continuously differentiable function could be exhibited with th...

onstrained optimization problems. Results for an industrial robot with six joints demonstrate that tailored optimization methods are very well suited for fast off-line optimization of robot trajectories. 1. Introduction In automotive industry, robotic manipulators play an important role for car production in automated assembly lines. The reduction of the cycle time of the production line is of great importance in order to reduce costs by improving efficiency. Here "optimal" robot trajectories and robust and efficient optimization methods for computing them are of great interest when planning production processes. 2. Modeling of the optimization problem Equations of Motion. Industrial robots have to act extremely fast. Therefore all dynamical effects have to be taken into account. The dynamical behavior of most industrial robots can be described in minimal coordinates by a system of second order differential equations (multibody system with tree

. The aim of the present paper is that of deriving a few unifying principles at the basis of numerically implementable existence conditions for systems of nonlinear inequalities in IR n . We define different criteria in terms of suitable merit functions and we derive, as special cases, most of the known regularity conditions employed for ensuring the convergence of algorithms towards feasible solutions. We establish also new extensions and connections with fixed point theory for nonlinear operators. Key words. Solution of nonlinear inequalities, feasible set, nonlinear programming. 1 Introduction The problem of determining a solution to a system of nonlinear inequalities is a fundamental problem in nonlinear optimization, which plays a major role both in global optimization and in constrained local optimization. In the general case, it is equivalent to a global optimization problem [10][8]. Indeed, the problem of determining ¯ x 2 IR n that satisfies a system of nonlinear inequa...