arXiv subjects

C. Yalçın Kaya

Publications and source records attributed to C. Yalçın Kaya.

24 records · Page 2Linked to original sources

arXiv2020

Steklov Convexification and a Trajectory Method for Global Optimization of Multivariate Quartic Polynomials

The Steklov function $μ_f(\cdot,t)$ is defined to average a continuous function $f$ at each point of its domain by using a window of size given by $t>0$. It has traditionally been used to approximate $f$ smoothly with small values of $t$. In this paper, we first find a concise and useful expression for $μ_f$ for the case when $f$ is a multivariate quartic polynomial. Then we show that, for large enough $t$, $μ_f(\cdot,t)$ is convex; in other words, $μ_f(\cdot,t)$ convexifies $f$. We provide an easy-to-compute formula for $t$ with which $μ_f$ convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving $μ_f$, a trajectory $x(t)$ emanating from the minimizer of the convexified $f$ and ending at $x(0)$, an estimate of the global minimizer of $f$. For a family of quartic polynomials, we provide an estimate for the size of a ball that contains all its global minimizers. Finally, we illustrate the working of our method by means of numerous computational examples.

math.OC↗

arXiv2020

Optimal control of the double integrator with minimum total variation

We study the well-known minimum-energy control of the double integrator, along with the simultaneous minimization of the total variation in the control variable. We derive the optimality conditions and obtain the unique optimal solution to the combined problem, where the initial and terminal boundary points are specified. We study the problem from a multi-objective optimal control viewpoint, constructing the Pareto front. We show that the unique asymptotic optimal control function, for the minimization of the total variation alone, is piecewise constant with one switching at the midpoint of the time horizon. For any instance of the boundary conditions of the problem, we prove that the asymptotic optimal total variation is exactly $2/3$ of the total variation of the minimum-energy control. We illustrate the results for a particular instance of the problem and include a link to a video which animates the solutions while moving along the Pareto front.

math.OC↗

arXiv2019

Markov-Dubins Interpolating Curves

A realistic generalization of the Markov--Dubins problem, which is concerned with finding the shortest planar curve of constrained curvature joining two points with prescribed tangents, is the requirement that the curve passes through a number of prescribed intermediate points/nodes. We refer to this generalization as the Markov--Dubins interpolation problem. We formulate this interpolation problem as an optimal control problem and obtain results about the structure of its solution using optimal control theory. The Markov--Dubins interpolants consist of a concatenation of circular ($C$) and straight-line ($S$) segments. Abnormal interpolating curves are shown to exist and characterized; however, if the interpolating curve contains a straight-line segment then it cannot be abnormal. We derive results about the stationarity, or criticality, of the feasible solutions of certain structure. In particular, any feasible interpolant with arc types of $CSC$ in each stage is proved to be stationary, i.e., critical. We propose a numerical method for computing Markov--Dubins interpolating paths. We illustrate the theory and the numerical approach by four qualitatively different examples.

math.OC↗

arXiv2018

Steklov Regularization and Trajectory Methods for Univariate Global Optimization

We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions. First, we show that the Steklov regularization convexifies a given univariate coercive function. Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function. For monic quartic polynomials, we prove that this trajectory does generate a global minimizer. In the process, we derive some properties of quartic polynomials. Comparisons are made with a previous approach which uses a quadratic regularization function. We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function.

math.OC↗

arXiv2018

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i)~Dykstra's algorithm, (ii)~the Douglas--Rachford (DR) method and (iii)~the Aragón Artacho--Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.

math.OC↗

arXiv2017

Markov-Dubins Path via Optimal Control Theory

Markov-Dubins path is the shortest planar curve joining two points with prescribed tangents, with a specified bound on its curvature. Its structure, as proved by Dubins in 1957 nearly 70 years after Markov posed the problem of finding it, is elegantly simple: a selection of at most three arcs are concatenated, each of which is either a circular arc of maximum (prescribed) curvature or a straight line. The Markov-Dubins problem and its variants have since been extensively studied in practical and theoretical settings. A reformulation of the Markov-Dubins problem as an optimal control problem was subsequently studied by various researchers using the Pontryagin maximum principle and additional techniques, to reproduce Dubins' result. In the present paper, we study the same reformulation, and apply the maximum principle, with new insights, to derive Dubins' result again. We prove that abnormal control solutions, which were ruled out in previous studies, do exist. We characterize them as a concatenation of at most two circular arcs and show that they are also solutions of the normal problem. More importantly, we prove that any feasible path of the types mentioned above is a stationary solution, i.e., that it satisfies the Pontryagin maximum principle. We propose a numerical method for computing Markov-Dubins path. We illustrate the theory and the numerical approach by three qualitatively different examples.

math.OC↗