On a Solving Bilevel D.C.-Convex Optimization Problems

Orlov, A. V.

doi:10.1007/978-3-030-58657-7_16

Cited by 4 publications

(3 citation statements)

References 35 publications

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“…In particular, it allows one to reformulate two-stage stochastic programming problems, whose second stage problem has DC (Difference-of-Convex) objective function and DC constraints, as equivalent unconstrained DC optimization problems and apply the well-developed apparatus of DC optimization to find their solutions (cf. analogous results for bilevel programming problems in [33,42]). Let us also note that exact penalty functions for singlestage stochastic programming were analysed in [24].…”

Section: Introductionmentioning

confidence: 75%

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Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming

Долгополик¹

2021

Preprint

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This work is devoted to an analysis of exact penalty functions and optimality conditions for nonsmooth two-stage stochastic programming problems. To this end, we first study the co-/quasi-differentiability of the expectation of nonsmooth random integrands and obtain explicit formulae for its co-and quasidifferential under some natural assumptions on the integrand. Then we analyse exact penalty functions for a variational reformulation of two-stage stochastic programming problems and obtain sufficient conditions for the global exactness of these functions with two different penalty terms. In the end of the paper, we combine our results on the co-/quasi-differentiability of the expectation of nonsmooth random integrands and exact penalty functions to derive optimality conditions for nonsmooth two-stage stochastic programming problems in terms of codifferentials.

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Section: Introductionmentioning

confidence: 75%

“…Thus, Theorem 3 opens a way for applications of DC programming algorithms to two-stage stochastic programming problems (cf. [33,42]).…”

Section: Exact Penalty Functionsmentioning

confidence: 99%

Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming

Долгополик¹

2021

Preprint

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“…There are two issues caused by the multipliers in the MPEC approach. Firstly, in theory, if the LL has more than one multiplier, the resulting MPEC is not equivalent to the original BLO if local optimality is considered [54]. Secondly, the introduced auxiliary multiplier variables limit the numerical efficiency in solving the BLO.…”

Section: Introductionmentioning

confidence: 99%

Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

Liu¹,

Gao²,

Zhang³

et al. 2021

Preprint

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Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community. BLO is able to handle problems with a hierarchical structure, involving two levels of optimization tasks, where one task is nested inside the other. In machine learning and computer vision fields, despite the different motivations and mechanisms, a lot of complex problems, such as hyper-parameter optimization, multi-task and meta-learning, neural architecture search, adversarial learning and deep reinforcement learning, actually all contain a series of closely related subproblms. In this paper, we first uniformly express these complex learning and vision problems from the perspective of BLO. Then we construct a value-function-based single-level reformulation and establish a unified algorithmic framework to understand and formulate mainstream gradient-based BLO methodologies, covering aspects ranging from fundamental automatic differentiation schemes to various accelerations, simplifications, extensions and their convergence and complexity properties. Last but not least, we discuss the potentials of our unified BLO framework for designing new algorithms and point out some promising directions for future research.

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On Solving Bilevel Optimization Problems with a Nonconvex Lower Level: The Case of a Bimatrix Game

Orlov

2021

Mathematical Optimization Theory and Operations Research

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On a Solving Bilevel D.C.-Convex Optimization Problems

Cited by 4 publications

References 35 publications

Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming

Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming

Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

On Solving Bilevel Optimization Problems with a Nonconvex Lower Level: The Case of a Bimatrix Game

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