A primal–dual algorithm for risk minimization

Kouri, Drew P.; Surowiec, Thomas M.

doi:10.1007/s10107-020-01608-9

Cited by 9 publications

(4 citation statements)

References 38 publications

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“…Deterministic optimal control problems with the Burgers' equation are studied, for example, in [19,76,77,78]. We refer the reader to [39,40,42,46] for risk-neutral and risk-averse control of the steady Burgers' equation.…”

Section: Distributed Control Of a Steady Burgers' Equationmentioning

confidence: 99%

Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

Milz¹

2022

Preprint

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We apply the sample average approximation (SAA) method to risk-neutral optimization problems governed by nonlinear partial differential equations (PDEs) with random inputs. We analyze the consistency of the SAA optimal values and SAA solutions. Our analysis exploits problem structure in PDEconstrained optimization problems, allowing us to construct deterministic, compact subsets of the feasible set that contain the solutions to the riskneutral problem and eventually those to the SAA problems. The construction is used to study the consistency using results established in the literature on stochastic programming. The assumptions of our framework are verified on three nonlinear optimization problems under uncertainty.

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Section: Distributed Control Of a Steady Burgers' Equationmentioning

confidence: 99%

Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

Milz¹

2022

Preprint

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“…The first one is the implicit approach where we remove the equality constraint c(y, u; 𝜔) = 0 via a control to solution map u  → y. 6,8,13 The second case is the full space approach, where we directly tackle the full problem (1) using the Lagrangian formulation. The latter formulation appears to be new in the context of risk-averse optimization.…”

Section: Introductionmentioning

confidence: 99%

“…Another solution strategy is to reformulate the problem and use interior‐point methods 8 . A duality‐based approach has also been recently proposed in Reference 13.…”

Section: Introductionmentioning

confidence: 99%

“…We consider two formulations of (1). The first one is the implicit approach where we remove the equality constraint

c\left(y,u;\omega \right)=0

via a control to solution map

u\mapsto y

6,8,13 . The second case is the full space approach, where we directly tackle the full problem (1) using the Lagrangian formulation.…”

Section: Introductionmentioning

confidence: 99%

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TTRISK: Tensor train decomposition algorithm for risk averse optimization

Antil

Dolgov

Onwunta

2022

Numerical Linear Algebra App

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This article develops a new algorithm named TTRISK to solve high‐dimensional risk‐averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so‐called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large‐scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID‐19. The results indicate that the risk‐averse framework is feasible with the tensor approximations under tens of random variables.

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Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

Milz

2023

Appl Math Optim

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A primal–dual algorithm for risk minimization

Cited by 9 publications

References 38 publications

Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

TTRISK: Tensor train decomposition algorithm for risk averse optimization

Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

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