Finite Optimization Problem Research Articles

Stability for the Erdős-Rothschild problem

Abstract Given a sequence $\boldsymbol {k} := (k_1,\ldots ,k_s)$ of natural numbers and a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ , such that, for every $c \in \{1,\dots ,s\}$ , the edges of colour c contain no clique of order $k_c$ . Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of nontrivial cases. In previous work with Pikhurko and Yilma, (Math. Proc. Cambridge Phil. Soc. 163 (2017), 341–356), we constructed a finite optimisation problem whose maximum is equal to the limit of $\log _2 F(n;\boldsymbol {k})/{n\choose 2}$ as n tends to infinity and proved a stability theorem for complete multipartite graphs G. In this paper, we provide a sufficient condition on $\boldsymbol {k}$ which guarantees a general stability theorem for any graph G, describing the asymptotic structure of G on n vertices with $F(G;\boldsymbol {k}) = F(n;\boldsymbol {k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$ . The proof uses a version of symmetrisation on edge-coloured weighted multigraphs.

Model Predictive Path Planning for an Autonomous Ground Vehicle in Rough Terrain

An autonomous ground vehicle (AGV) in rough terrain typically experiences uncertain environment. Because the uncertainty makes overall performance of autonomous navigation degrade, the AGV requires a suitable path to maintain or improve the performance against the uncertainty. In order to handle this problem, this study proposes a model predictive path planning algorithm by employing a passivity-based model predictive control (MPC) optimization setup. The model predictive path planning method is formulated as a finite optimization problem with an objective function and several constraints. In the cost function, environment perception result about the AGV’s own neighborhood is included and the only traversable region has low cost value. To reflect dynamic characteristics of the AGV, the proposed method utilizes dynamic and kinematic models of the AGV as equality constraints and limited range of states and control input as inequality constraints. In addition, the stability of the path planning method is improved by a passivity constraint. The solution of the optimization problem is obtained using the particle swarm optimization (PSO) method. Finally, field tests are conducted to validate the performance of the proposed algorithm, and satisfactory results of the autonomous navigation were obtained.

The problem of placing a two-stage production with restrictions on the capacity of the first stage enterprises

Facility location problems are a fertile ground for the development of new modelling techniques, innovative solution algorithms and exciting applications. The article describes a problem of placing a two-stage production with restrictions on the capacity of the first stage enterprises. Such problems arise, for example, in strategic planning of development of the region. The problems of optimal placement companies and identifying areas of their influence are interesting for business (allocation of warehouses, shops, service outlets, etc.) and public companies (schools, hospitals, fire stations and so forth.) The tasks of research were to construct the mathematical model for a two-stage of optimal location-allocation problem considering the restrictions on the capacity of enterprises of the first stage; to describe the solution method and to formulate an algorithm. In this paper, authors gave the mathematical model of the problem, a brief description of the solution method and algorithm. Aggregate cost of product delivery was chosen as a criterion for optimal location problem. Solving methods are based on principles of infinite-dimensional optimization and duality theory. The approach to solution of this type of problem is based on the solution of the problem of optimal partitioning set and discrete multi-stage location problem. A unified approach to solving optimal partitioning set problems lies in the conversion of the initial problems into infinite-dimensional mathematical programming problems by means of the characteristic functions, and then into the finite optimization problem using Lagrangian functional. Iterative algorithm to solve the problem has been elaborated. The algorithm combines a method of potentials being applied for classical problem of linear programming of transportation type and N.Z. Shor’s algorithm making it possible to solve optimization problem of nonsmooth function. Software product has been developed to solve two-stage problems of optimal location of enterprises with continuously resources. The results obtained by authors make it possible to solve a range of practical problems connected with strategic planning in the sphere of production, social and economic activities.

Robust Control Strategy for a Conduction–Convection System Based on the Scenario Optimization

This paper deals with the robust control of an uncertain conduction–convection system in the framework of probabilistic control design based on both the geometric control and the scenario optimization approach. Thus, a robust control strategy that copes with parameter uncertainties is proposed for a heated rod taken as an application example of a conduction–convection system. The design approach consists in two steps. In the first step, assuming a nominal model, a state feedback that yields a stable linear lumped parameter system, of first order, in closed loop is designed by means of geometric control theory. The stability of the resulting closed-loop system is demonstrated based on the perturbation theorem from semigroup theory. The second step consists in defining the input reference of the designed state feedback by a structured robust controller. The parameter tuning of the structured controller is formulated as a semi-infinite (or robust) optimization problem which is, then, relaxed using the scenario approach leading to a standard finite optimization problem. The solution of this scenario optimization problem is achieved using a genetic algorithm. The proposed control strategy is adopted to cope with parameter uncertainties in the problem of heating a steel rod. The effectiveness of the proposed robust control strategy is demonstrated by simulation.

The Erdős–Rothschild problem on edge-colourings with forbidden monochromatic cliques

AbstractLet k := (k1,. . .,ks) be a sequence of natural numbers. For a graph G, let F(G;k) denote the number of colourings of the edges of G with colours 1,. . .,s such that, for every c ∈ {1,. . .,s}, the edges of colour c contain no clique of order kc. Write F(n;k) to denote the maximum of F(G;k) over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases.We prove that, for every k and n, there is a complete multipartite graph G on n vertices with F(G;k) = F(n;k). Also, for every k we construct a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/${n\choose 2}$ as n tends to infinity. Our final result is a stability theorem for complete multipartite graphs G, describing the asymptotic structure of such G with F(G;k) = F(n;k) · 2o(n2) in terms of solutions to the optimisation problem.

Numerical solution of optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions

In this work we consider optimization problems for processes described by semi-linear partial differential equations of elliptic type with discontinuous coefficients and solutions (with imperfect contact matching conditions), with controls involved in the coefficients. Finite difference approximations of optimization problems are constructed. For the numerical implementation of finite optimization problems differentiability and Lipshitz-continuity of the grid functional of the approximating grid problems are proved. An iterative method for solving boundary value problems of contact for PDEs of elliptic type with discontinuous coefficients and solutions is developed and validated. The convergence of the iterative process is investigated. And the convergence rate of iterations (with calculated constants) is estimated.

Open loop optimal downhill driving brake strategies using non-linear programming

In this paper open loop optimal trajectories for downhill driving are described. Problem formulation including process modelling, model simplification, transformation into a finite optimisation problem, and implementation into the TOMLAB optimisation package are presented. Results show that there is a large potential in controlling the complete brake system of a heavy duty truck and thereby simultaneously improve both mean speed (transport efficiency) and component wear cost. The resulting optimal trajectories define the upper limit for what is theoretically achievable in a real, closed loop, controller implementation and can be used to both inspire and verify the development of such algorithms.

Finite-element approximation of 2D elliptic optimal design

We consider a problem of elliptic optimal design in two space dimensions. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n = 2 , Šveràk [V. Šveràk, On optimal shape design, J. Math. Pures Appl. 72 (1993) 537–551] proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complementary sets have a uniformly bounded number of connected components. The proof in [V. Šveràk, On optimal shape design, J. Math. Pures Appl. 72 (1993) 537–551] is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology H c and the continuous dependence of the solutions of the Dirichlet Laplacian in H 1 with respect to it. In this article we introduce a finite-element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. The discrete optimal design problem admits at least one solution since it is a finite optimization problem. We prove that any limit in H c of discrete optimal shapes, when the mesh-size tends to zero, is an optimal domain for the continuous optimal design problem. The proof relies on the following two key facts: (a) any open bounded set of R 2 can be approximated in H c by a sequence of triangulated domains, (b) finite-element approximations of the Dirichlet Laplacian in the triangulated domains converge in H 1 to the solutions of the continuous Dirichlet problem whenever the triangulated domains converge in H c .

Solving Semi-Infinite Optimization Problems with Interior Point Techniques

We introduce a new numerical solution method for semi-infinite optimization problems with convex lower level problems. The method is based on a reformulation of the semi-infinite problem as a Stackelberg game and the use of regularized nonlinear complementarity problem functions. This approach leads to central path conditions for the lower level problems, where for a given path parameter a smooth nonlinear finite optimization problem has to be solved. The solution of the semi-infinite optimization problem then amounts to driving the path parameter to zero. We show convergence properties of the method and give a number of numerical examples from design centering and from robust optimization, where actually so-called generalized semi-infinite optimization problems are solved. The presented method is easy to implement, and in our examples it works well for dimensions of the semi-infinite index set at least up to 150.

Solving fuzzy variational inequalities over a compact set

This paper studies the fuzzy variational inequalities over a compact set. By using the tolerance approach, we show that solving such problems can be reduced to a semi-infinite programming problem. A relaxed cutting plane algorithm is proposed. In each iteration, we solve a finite optimization problem and add one more constraint. The proposed algorithm chooses a point at which the infinite constraints are violated to a degree rather than at which the violation is maximized. The iterative process ends when an optimal solution is identified. A convergence proof, under some mild conditions, is given. An efficient implementation based on the “entropic regularization” techniques is also included. To illustrate the solution procedure, a numerical example is provided.

Metric regularity of the feasible set mapping in semi-infinite optimization

For parametric systems of (finitely many) equations and (infinitely many) inequalities the well-known concept of metric regularity is shown to be equivalent to the so-called extended Mangasarian-Fromovitz constraint qualification. By this, a corresponding result obtained by Robinson for finite optimization problems my be transferred to semi-infinite optimization. For the proof a local epigraph representation of the constraint set is mainly used.