Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Control Of Nonlinear Dynamical Systems: Methods And Applications

Abstract: Optimal control.- Method of decomposition (the first approach).- Method of decomposition (the second approach).- Stability based control for Lagrangian mechanical systems.- Piecewise linear control for mechanical systems under uncertainty.- Continuous feedback control for mechanical systems under uncertainty.- Control in distributed-parameter systems.- Control system under complex constraints.- Optimal control problems under complex constraints.- Time-optimal swing-up and damping feedback controls of a nonlinear pendulum.

Robust optimization of sums of piecewise linear functions with application to inventory problems

Abstract: Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be “robustified” is not concave (or linear) with respect to the perturbing parameters. In this paper, we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semi-definite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multi-period inventory problems that is exact whether the uncertainty region reduces to a L1-norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem.

Acceleration of Univariate Global Optimization Algorithms Working with Lipschitz Functions and Lipschitz First Derivatives

Abstract: This paper deals with two kinds of the one-dimensional global optimization problem over a closed finite interval: (i) the objective function $f(x)$ satisfies the Lipschitz condition with a constant $L$; (ii) the first derivative of $f(x)$ satisfies the Lipschitz condition with a constant $M$. In the paper, six algorithms are presented for the case (i) and six algorithms for the case (ii). In both cases, auxiliary functions are constructed and adaptively improved during the search. In the case (i), piecewise linear functions are constructed and in the case (ii) smooth piecewise quadratic functions are used. The constants $L$ and $M$ either are taken as values known a priori or are dynamically estimated during the search. A recent technique that adaptively estimates the local Lipschitz constants over different zones of the search region is used to accelerate the search. A new technique called the local improvement is introduced in order to accelerate the search in both cases (i) and (ii). The algorithms are...

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

Abstract: A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly.