Solution Existence for Time-Varying Infinite Horizon Quadratic Programming

Schochetman, Irwin E.; Smith, Robert L.; Tsui, Sze-Kai

doi:10.1006/jmaa.1995.1347

Cited by 5 publications

(7 citation statements)

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“…Note that even if K + N , i.e., P (F ), is closed in K ⊥ , (P (L)) need not admit an optimal solution-even though R is positive definite. (See [6] for a counter-example. )…”

Section: An Applicationmentioning

confidence: 95%

“…Note that R is a positive definite, bounded linear operator on K ⊥ which need not be strictly positive definite. It will be if the positive spectrum of Q is bounded away from 0 [7]. Note also that if x = η ⊕ ξ uniquely, for x ∈ H , η ∈ K, ξ ∈ K ⊥ , then…”

Section: An Applicationmentioning

confidence: 98%

“…As in [7], solving (P (L)) is equivalent to solving (L) in the following sense. If ξ ∈ P (F ) is optimal for (P (L)), i.e., ξ ∈ P (F ) * , then there exists η ∈ K (not necessarily unique) such that x = η ⊕ ξ is in F , and is necessarily optimal for (L) since x, Q(x) = ξ, R(ξ ) .…”

Section: An Applicationmentioning

confidence: 99%

“…In Section 3, we give an application of our main results to an infinite-dimensional, positive semi-definite, linear quadratic programming problem (as in [6,7]). Specifically, under appropriate conditions, we show (Theorem 3.2) that optimal solutions exist.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

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Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

Schochetman

Smith

Tsui

2007

Journal of Mathematical Analysis and Applications

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We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.

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“…Note that even if K + N , i.e., P (F ), is closed in K ⊥ , (P (L)) need not admit an optimal solution-even though R is positive definite. (See [6] for a counter-example. )…”

Section: An Applicationmentioning

confidence: 95%

Section: An Applicationmentioning

confidence: 98%

Section: An Applicationmentioning

confidence: 99%

Section: Introduction and Problem Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

Schochetman

Smith

Tsui

2007

Journal of Mathematical Analysis and Applications

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“…See [4] for an example. In this section, we establish sufficient conditions for (3.1) to admit an optimal solution.…”

Section: Y | : (X Y) ∈ S(k N) For S(k N) ≠ ∅ −∞ For S(k N) = mentioning

confidence: 99%

On the closure of the sum of closed subspaces

Schochetman

Smith

Tsui

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators.2000 Mathematics Subject Classification. 46C05, 90C20.

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Some Stability Properties of Parametric Quadratically Constrained Nonconvex Quadratic Programs in Hilbert Spaces

Dong¹

2017

Acta Math Vietnam

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Stability of nonconvex quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces is investigated. We present several stability properties of the global solution map, and the continuity of the optimal value function, assuming that the problem data undergoes small perturbations.Mathematics Subject Classification (2010). 90C20, 90C30, 90C31. Key Words. quadratic program in Hilbert spaces, convex quadratic constraints, solution existence, Legendre form, recession cone, solution set, solution map, optimal value function. * V. V. Dong,

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Solution Existence for Time-Varying Infinite Horizon Quadratic Programming

Cited by 5 publications

References 0 publications

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

On the closure of the sum of closed subspaces

Some Stability Properties of Parametric Quadratically Constrained Nonconvex Quadratic Programs in Hilbert Spaces

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