Algorithmic differentiation for piecewise smooth functions: a case study for robust optimization

Fiege, Sabrina; Walther, Andrea; Kulshreshtha, Kshitij; Griewank, Andreas

doi:10.1080/10556788.2017.1333613

Cited by 10 publications

(5 citation statements)

References 10 publications

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“…Here, we provide examples of nonlinear robust optimization problems considered in the literature. We emphasize cases with infinite-cardinality uncertainty sets, but we note that finite-cardinality uncertainty set examples are also commonplace; see, for example, the minimax regret and test problems in Fiege et al (2018), Hare and Macklem (2013), and Hare and Nutini (2013). The illustrative examples in Section 3.2 are used in later sections to demonstrate solution techniques and available software.…”

Section: Applications and Illustrative Examplesmentioning

confidence: 99%

A survey of nonlinear robust optimization

Leyffer

Menickelly

Munson

et al. 2020

INFOR: Information Systems and Operational Research

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Robust optimization (RO) has attracted much attention from the optimization community over the past decade. RO is dedicated to solving optimization problems subject to uncertainty: design constraints must be satisfied for all the values of the uncertain parameters within a given uncertainty set. Uncertainty sets may be modeled as deterministic sets (boxes, polyhedra, ellipsoids), in which case the RO problem may be reformulated via worst-case analysis, or as families of distributions. The challenge of RO is to reformulate or approximate robust constraints so that the uncertain optimization problem is transformed into a tractable deterministic optimization problem. Most reformulation methods assume linearity of the robust constraints or uncertainty sets of favorable shape, which represents only a fraction of real-world applications. This survey addresses nonlinear RO and includes problem formulations and applications, solution approaches, and available software with code samples.

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Section: Applications and Illustrative Examplesmentioning

confidence: 99%

A survey of nonlinear robust optimization

Leyffer

Menickelly

Munson

et al. 2020

INFOR: Information Systems and Operational Research

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show abstract

“…Obviously, (28) involves ś m i"1 k i and Proof. We will consider the representations (27) from which (26) can be directly obtained in the form (28). Firstly, the independent variables x j are linear functions of themselves with gradient a " e j and inhomogeneity α " 0.…”

Section: The Two-term Polyhedral Decompositionmentioning

confidence: 99%

“…Finally, one should always keep in mind that the task of minimizing a piecewise linear function will most likely occur as an inner problem in the optimization of a piecewise smooth and nonlinear function. As we have shown in [27] the local piecewise linear model problem can be obtained easily by a slight generalization of automatic or algorithmic differentiation, e.g., ADOL-C [28] and Tapenade [29]. as LIKQ in [7] and obviously requires that no more than n switches are active atx.…”

mentioning

confidence: 99%

Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Griewank

Walther

2020

Algorithms

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For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f and a concave part f ^ , including a pair of generalized gradients g ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g and − g ^ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization.

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“…The theory of analytical differentiation has been fruitfully applied to many areas of computer science, like deep learning [Baydin et al 2017] (choosing weights in neural nets by gradient descent optimisation methods), algorithmic differentiation [Fiege et al 2018; (providing techniques for efficient differentiation via source code transformation) and, recently, programming languages with first-class differentiation have begun to appear, see e.g. [Walter and Lehmann 2013].…”

Section: Introductionmentioning

confidence: 99%

Taylor subsumes Scott, Berry, Kahn and Plotkin

Barbarossa

Manzonetto

2019

Proc. ACM Program. Lang.

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The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential λ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in λ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity.

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Algorithmic differentiation for piecewise smooth functions: a case study for robust optimization

Cited by 10 publications

References 10 publications

A survey of nonlinear robust optimization

A survey of nonlinear robust optimization

Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Taylor subsumes Scott, Berry, Kahn and Plotkin

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