A. Barclay, P. Gill, and J. Rosen. Sqp methods and their application to numerical optimal control, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Section 3. Academic test problems illustrating these ideas appear in Section 5. These test examples will show that the order estimates, even though they are sharp, do not tell the full story. Section 6 will discuss these di#erences. Related discussions that address di#erent issues can be found in [1,8,9,15,16]. Finally, this paper is part of a larger investigation concerning the optimization of machine tool paths. In this application high precision is necessary. This is re#ected in our choice of stopping tolerances. 2. SOCS Sparse optimal control software (SOCS) is a software package developed at ....

....collocation form. However, it has some redundant variables. Note that U # 1 ;U # 2 appear only in the expression u n 1 (h n 1 =2) U # 1 U # 2 ) which is equal to u n . This does not alter our observations but it could impact on actually using the IRK formulations. Similar comments hold for HS [1]. SOCS implements HS as a collocation and not directly as an IRK. It is known that DAEs pose problems for IRK methods and a reduction of the order of convergence may occur. Most of the existing theory deals with DAEs in Hessenberg form. For index two Hessenberg DAEs y # = f(y; u) 10a) 0=g(y) ....

A. Barclay, P.E. Gill, J. Ben Rosen, SQP methods and their application to numerical optimal control, Report NA 97-3, Department of Mathematics, University of California, San Diego, 1997.

.... functions, see Zefran and Kumar [17] as they lead to simple expressions for magnitude and rate constraints (and are common choice in nite element methods) A more re ned choice would be piecewise cubic Hermite polynomials, that are quite successful in classic optimal control settings [1]. More general base functions can be considered for unilateral or quantized control inputs. In the context of real time applications, the computation of the tensors # ; # can be performed o line through FFT or other techniques. 4.1 Existence of local trajectories In general no analytic ....

A. Barclay,P. E. Gill, and J. B. Rosen. SQP methods and their application to numerical optimal control. Technical report, University of California, San Diego, February 1997.

....grid points. NLPs can be solved most efficiently numerically by SQP methods. In each SQP iteration a current guess of the solution y is improved by the solution of a quadratic subproblem derived from a quadratic approximation of the Lagrangian of the NLP subject to the linearized constraints [3, 8]. The NLPs resulting from a direct collocation discretization have several special properties [15] The NLPs are of large scale with very many variables and very many constraints. Most of the NLP constraints are active at the solution, e.g. the equality constraints from collocation. Thus, the ....

A. Barclay, P.E. Gill, J.B. Rosen. SQP methods and their application to numerical optimal control. In W.H. Schmidt, K. Heier, L. Bittner, R. Bulirsch (eds.): Variational Calculus, Optimal Control and Applications, International Series of Numerical Mathematics 124:207-- 222, Birkhauser, Basel, 1998.

....small system with tridiagonal system has to be solved. This is typical for many applications, in particular those in dynamical systems. Many SQP based codes for optimal control problems governed by ODEs or DAEs exploit this structure efficiently in their numerical linear algebra. See, e.g. [1], 2] 42] 58] 62] and the references therein. For many applications, in particular those governed by PDEs, such factorizations of the Jacobian J(x) of C(x) are not feasible from a practical point of view, but solution techniques for C y (y; u)v y = r and C y (y; u) T v y = r are available. ....

....problems, many algorithms follow this approach. Often, projection techniques are used to handle the box constraints, see e.g. 28] 51] Recently, so called all at once approaches that treat both y and u as independent variables have been proposed to solve optimal control problems, see e.g. [1], 2] 4] 29] 32] 33] 34] 35] 36] 37] 39] 41] 42] 57] 58] 62] Since they move towards optimality and feasibility at the same time, they offer significant advantages. SQP methods are of particular interest. They do not require the possibly very expensive solution of the ....

A. Barclay, P. E. Gill, and J. B. Rosen, SQP methods and their application to numerical optimal control, Numerical Analysis Report 97--3, Department of Mathematics, University of California, San Diego, La Jolla, CA, 1997.

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A. Barclay, P. Gill, and J. Rosen. Sqp methods and their application to numerical optimal control, 1997.

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A. Barclay, P.E. Gill, and J. B. Rosen, SQP methods and their applications to numerical optimal control, to appear.

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