Piecewise affine functions and polyhedral sets<sup>∗</sup>

Gorokhovik, Valentin V.; Zorko, O. I.; Birkhoff, Garrett

doi:10.1080/02331939408844018

Cited by 36 publications

(28 citation statements)

References 9 publications

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“…2). The regression function is m 8 , m 9 , m 10 or m 11 ods outperform our estimate for m 9 (which is a very simple function depending in fact only of two of the components of the predictor variable), but our estimate clearly outperforms the other estimates in case of l = 5000 and m 10 and for l ∈ {500, 5000} in case of m 11 .…”

Section: Figcontrasting

confidence: 43%

“…Since any continuous piecewise linear function can be represented as a max-min of finite number of linear functions (see [5,11]) we consider maxima of minima of linear functions. More precisely, let K l ∈ N and L 1,l , .…”

Section: Definition Of the Estimatementioning

confidence: 99%

“…Since each continuous piecewise linear function can be represented as a maxima of minima of linear functions [5,11], we use such a representation to estimate regression functions. We fit a linear spline function with free knots to the data.…”

Section: Introductionmentioning

confidence: 99%

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An algorithm for the estimation of a regression function by continuous piecewise linear functions

2008

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Section: Figcontrasting

confidence: 43%

Section: Definition Of the Estimatementioning

confidence: 99%

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An algorithm for the estimation of a regression function by continuous piecewise linear functions

2008

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“…in the small note [29]. The term 'convex component' was introduced by Gorokhovik and Zorko in the paper [30] devoted to nonconvex polyhedral sets.…”

Section: Convex Components and Convex Complements Of A Setmentioning

confidence: 99%

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

Gorokhovik

2018

Optimization

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The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to more larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the DemyanovRubinov subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions and the Dini-Hadamard (directional) subdifferential of directionally differentiable functions. Some applications of Demyanov-Rubinov subdifferentials to extremal problems are considered.

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“…Furthermore, it has been shown in [4]- [6] that MMPS systems are equivalent to a particular class of hybrid systems, namely continuous piecewise affine (PWA) systems. The relation between PWA and MMPS systems is useful for the investigation of structural properties of PWA systems such as observability and controllability but also in designing controller schemes like model predictive control (MPC) [7], [8].…”

Section: Introductionmentioning

confidence: 99%

Model predictive control for stochastic max-min-plus-scaling systems - an approximation approach

Farahani

Boom

Schutter

2011

IEEE Conference on Decision and Control and European Control Conference

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If you want to cite this report, please use the following reference instead:S.S. Farahani, T. van den Boom, and B. De Schutter, "Model predictive control for stochastic max-min-plus-scaling systems -An approximation approach," Proceedings of the 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, Florida, pp. 391-396, Dec. 2011 Abstract-A large class of discrete-event and hybrid systems can be described by a max-min-plus-scaling (MMPS) model, i.e., a model in which the main operations are maximization, minimization, addition, and scalar multiplication. Further, Model Predictive Control (MPC), which is one of the most widely used advanced control design methods in the process industry due to its ability to handle constraints on both inputs and outputs, has already been extended to both deterministic and stochastic MMPS systems. However, in order to compute an MPC controller for a general MMPS system, a nonlinear, nonconvex optimization problem has to be solved. In addition, for stochastic MMPS systems, the problem is computationally highly complex since the cost function is defined as the expected value of an MMPS function and its evaluation leads to a complex numerical integration. The aim of this paper is to decrease this computational complexity by applying an approximation method that is based on the raw moments of a random variable, to a stochastic MMPS system with a Gaussian noise. In this way, the problem can be transformed into a sequence of convex optimization problems, providing that linear or convex MPC input constraints are considered.

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Piecewise affine functions and polyhedral sets^∗

Cited by 36 publications

References 9 publications

An algorithm for the estimation of a regression function by continuous piecewise linear functions

An algorithm for the estimation of a regression function by continuous piecewise linear functions

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

Model predictive control for stochastic max-min-plus-scaling systems - an approximation approach

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Piecewise affine functions and polyhedral sets∗

Cited by 36 publications

References 9 publications

An algorithm for the estimation of a regression function by continuous piecewise linear functions

An algorithm for the estimation of a regression function by continuous piecewise linear functions

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

Model predictive control for stochastic max-min-plus-scaling systems - an approximation approach

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Piecewise affine functions and polyhedral sets^∗