On tridiagonal linear complementarity problems

Schmidt, Jochen

doi:10.1007/bf01399692

Cited by 10 publications

(9 citation statements)

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“…Moreover, this procedure turns out to be effective in tridiagonal linear complementarity and in discrete obstacle problems, see Schmidt [,15], [,16], Kriitzschmar [9]. Now, this general strategy is shown to apply also to the problem (1.1), (1.2) of convex approximation with cubic Cl-splines.…”

Section: Fw" } F(s)= I F ~ S"(x)2 Dx-t-pi_l[s(xi_l)--gi_l]2-1cpi[s(ximentioning

confidence: 95%

An unconstrained dual program for computing convexC 1-spline approximants

Schmidt

1987

Computing

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--ZusammenfassungAn Unconstrained Dual Program for Computing Convex C1-Spline Approximants. In the present paper the problem of approximating given data sets by convex cubic CX-splines is considered. To this programming problem a dual program is constructed which is unconstrained. Therefore an efficient computational treatment is possible. AMS (MOS)SubjectClassifications: 65D 10, 41A15, 90C20. Key words: Specia~y structured optimization problem, Fenchel dualization, return-formula.. Eine dnale Optimierongsaufgabe ohne Nebenbedingungen zum Problem der konvexen Approximationdutch C1-Splines. Es wird die Aufgabe betrachtet, eine vorgegebene Datenmenge dutch konvexe kubische C~-Splines zu approximierert. Zu diesem Problem wird eine zugeh6rige duale Optimierungsaufgabe konstruiert, welche den Vorteil hat, dab keine Nebenbedingungen auftreten, und welche sich daher effektiver ats ~tas Ausgangsproblem numerisch behandeln l~iBt.

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Section: Fw" } F(s)= I F ~ S"(x)2 Dx-t-pi_l[s(xi_l)--gi_l]2-1cpi[s(ximentioning

confidence: 95%

An unconstrained dual program for computing convexC 1-spline approximants

Schmidt

1987

Computing

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“…This can be done by applying a dualization concept introduced by Burmeister/HeB/Schmidt [1] and Dietze/Schmidt [3] and used in further papers, e.g. in [11,12,13]. As it will be shown, a dual program associated with the present primal program (1.3) is Program DA (1.5)…”

Section: Program Pamentioning

confidence: 96%

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Schmidt

Scholz²

1990

Numer. Math.

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Summary. In this paper the problem of smoothing a given data set by cubic C 2-splines is discussed. The spline may required to be convex in some parts of the domain and concave in other parts. Application of splines has the advantage that the smoothing problem is easily discretized. Moreover, the special structure of the arising finite dimensional convex program allows a dualization such that the resulting concave dual program is unconstrained. Therefore the latter program is treated numerically much more easier than the original program. Further, the validity of a return-formula is of importance by which a minimizer of the original program is obtained from a maximizer of the dual program.The theoretical background of this general approach is discussed and, above all, the details for applying the strategy to the present smoothing problem are elaborated. Also some numerical tests are presented.

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“…The problems considered in [1,2] and [8] are special cases of (1.5) with M..={1, 2 ..... n+ 1} and P,={(i, i+ 1): i,=l(1)n}. Another problem of interest is the obstacle problem in IR".…”

Section: Introductionmentioning

confidence: 99%

“…3. Whereas the transformations of (1.10) into (1.5) considered in [8] are restricted to tridiagonal matrices A.-=A n the approach presented here is applicable to quadratic minimization problems (I.10) where the matrix A is only assumed to be a H-matrix. Finally, two concrete examples of discretized obstacle problems and numerical tests are considered in Chap.…”

Section: Introductionmentioning

confidence: 99%

“…In [1,2,8] constrained convex minimization problems in convex spline interpolations and in tridiagonal complementarity problems have been transformed into unconstrained convex minimization problems by Fenchel-Rockafellar dualization. The applicability of the decomposition-dualization technique considered in [1,2,8] is restricted to those constrained quadratic minimization problems where the Hesse matrix of the convex objective functional is tridiagonal. The aim of our investigations is to show that also more general constrained convex minimization problems can be effectively solved by decomposition and Fenchel-Rockafellar dualization, for instance problems in quadratic programming where the Hesse matrix of the objective functional is a H-matrix, e.g.…”

Section: Introductionmentioning

confidence: 99%

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A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Krätzschmar

1989

Numer. Math.

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Summary.In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated by H-matrices. Numerical tests for solving obstacle problems in lR x discretized by using piecewise quadratic finite elements and in IR 2 by using the five-point difference approximation are presented.

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On tridiagonal linear complementarity problems

Cited by 10 publications

References 10 publications

An unconstrained dual program for computing convexC 1-spline approximants

An unconstrained dual program for computing convexC 1-spline approximants

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

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