Hölder continuity of perturbed solution set for convex optimization problems

Li, X.B.; Li, S.J.

doi:10.1016/j.amc.2014.01.095

Cited by 11 publications

(7 citation statements)

References 26 publications

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“…For parametrized convex programs, [19] establishes continuity properties of the optimal value as a function of certain specific perturbations. Reference [29] showed Hölder continuity of the optimal solution with respect to the parameters in strongly convex programs with parametrized constraint sets satisfying certain properties. For uniquely solvable linear complementarity problems, [32] characterized a Lipschitz constant for the continuity of the unique optimal solution (again, as a function of the parameters).…”

Section: Prior Workmentioning

confidence: 99%

An Algorithm to Warm Start Perturbed (WASP) Constrained Dynamic Programs

Gupta¹,

Deshpande²,

Canova³

2021

Preprint

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Receding horizon optimal control problems compute the solution at each time step to operate the system on a near-optimal path. However, in many practical cases, the boundary conditions, such as external inputs, constraint equations, or the objective function, vary only marginally from one time step to the next. In this case, recomputing the optimal solution at each time represents a significant burden for real-time applications. This paper proposes a novel algorithm to approximately solve a perturbed constrained dynamic program that significantly improves the computational burden when the objective function and the constraints are perturbed slightly. The method hinges on determining closed-form expressions for first-order perturbations in the optimal strategy and the Lagrange multipliers of the perturbed constrained dynamic programming problem are obtained. This information can be used to initialize any algorithm (such as the method of Lagrange multipliers, or the augmented Lagrangian method) to solve the perturbed dynamic programming problem with minimal computational resources.

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Section: Prior Workmentioning

confidence: 99%

An Algorithm to Warm Start Perturbed (WASP) Constrained Dynamic Programs

Gupta¹,

Deshpande²,

Canova³

2021

Preprint

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“…Since κ(x) is always nonnegative, it restores feasibility by enlarging the range of admissible constraint function values. Indeed, the point d where a minimum is reached in the optimization problem in (7) is easily seen to be always feasible for (P x ). Moreover, for every x ∈ K, the following relations hold thanks to Assumption A: (10) and…”

Section: Main Properties Of Subproblem (P X )mentioning

confidence: 99%

“…In turn, by [10, Theorem 3.2], for every x ∈ K, the set-valued mapping X has the Aubin property relative to K at x for any element belonging to X (x) (see [11] for the definition of the Aubin property). Even more, in the light of [11,Theorems 9.38 and 9.30] being X outer semicontinuous and locally bounded at x relative to K, for every x ∈ K, X is Lipschitz continuous (see [11] for the definition of the Lipschitz continuity in the context of set-valued mappings) on a neighborhood of x relative to K. Therefore, in view of [7,Theorem 3.3], for every x ∈ K, there exist μ > 0 and a neighborhood V of x such that, for every y, z…”

Section: Assumption Bmentioning

confidence: 99%

Diminishing Stepsize Methods for Nonconvex Composite Problems via Ghost Penalties: from the General to the Convex Regular Constrained Case

Facchineia¹,

Kungurtsevb²,

Lampariello³

et al. 2020

Preprint

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In this paper we first extend the diminishing stepsize method for nonconvex constrained problems presented in [4] to deal with equality constraints and a nonsmooth objective function of composite type. We then consider the particular case in which the constraints are convex and satisfy a standard constraint qualification and show that in this setting the algorithm can be considerably simplified, reducing the computational burden of each iteration.

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“…□ Remark 4.2 Recently, there have been many papers devoted to the Hölder continuity for optimization problems [5,9,35]. However, since the imposed assumptions relate to strong monotonicity/convexity, the solution sets are unique.…”

Section: Vector Optimization Problemsmentioning

confidence: 99%

On Hölder continuity of approximate solution maps to vector equilibrium problems

Anh¹,

Nguyen²,

Tam³

2017

Turk J Math

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In this article, we consider parametric vector equilibrium problems in normed spaces. Sufficient conditions for Hölder continuity of approximate solution mappings where they are set-valued are established. As applications of these results, the Hölder continuity of the approximate solution mappings for vector optimization problems and vector variational inequalities are derived at the end of the paper. Our results are new and include the existing ones in the literature.

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Hölder continuity of perturbed solution set for convex optimization problems

Cited by 11 publications

References 26 publications

An Algorithm to Warm Start Perturbed (WASP) Constrained Dynamic Programs

An Algorithm to Warm Start Perturbed (WASP) Constrained Dynamic Programs

Diminishing Stepsize Methods for Nonconvex Composite Problems via Ghost Penalties: from the General to the Convex Regular Constrained Case

On Hölder continuity of approximate solution maps to vector equilibrium problems

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