Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints

We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into accountthe multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated

On the Approximation of Unbounded Convex Sets by Polyhedra

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the connections between existing results before introducing a new notion of polyhedral approximation called ($\varepsilon,\delta$)-approximation that integrates the unbounded case in a meaningful way. Some basic results about ($\varepsilon,\delta$)- approximations are proven for general convex sets. In the last section an algorithm for the computation of ($\varepsilon,\delta$)-approximations of spectrahedra is presented. Correctness and finiteness of the algorithm are proven.Comment: 22 pages, 4 figures, 1 tabl

Some outer approximation methods for semi-infinite optimization problems

AbstractThe paper starts with a simple model and convergence theorem for outer approximation methods. This general framework is used to unifyingly derive and modify certain exchange methods, cutting methods and discretization methods for semi-infinite programming problems. By that, in particular, a cutting plane method for convex semi-infinite programs is developed. For a practically reasonable specification (the method is more generally stated), the subproblems in the given algorithm are moderately sized quadratic problems, and each step of the algorithm can be performed by means of finitely many operations

Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems

In this paper, we generalize the extended supporting hyperplane algorithm for a convex continuously differentiable mixed-integer nonlinear programming problem to solve a wider class of nonsmooth problems. The generalization is made by using the subgradients of the Clarke subdifferential instead of gradients. Consequently, all the functions in the problems are assumed to be locally Lipschitz continuous. The algorithm is shown to converge to a global minimum of an MINLP problem if the objective function is convex and the constraint functions are f degrees-pseudoconvex. With some additional assumptions, the constraint functions may be f degrees-quasiconvex

Robustness of systems with uncertainties in the input

In B. R. Barmish (IEEE Trans. Automat. ControlAC-22, No. 7 (1977) 123, 124; AC-24, No. 6 (1979), 921-926) and B. R. Barmish and Y. H. Lin ("Proceedings of the 7th IFAC World Congress, Helsinki 1978") a new notion of "robustness" was defined for a class of dynamical systems having uncertainty in the input-output relationship. This paper generalizes the results in the above-mentioned references in two fundamental ways: (i) We make significantly less restrictive hypotheses about the manner in which the uncertain parameters enter the system model. Unlike the multiplicative structure assumed in previous work, we study a far more general class of nonlinear integral flows, (ii) We remove the restriction that the admissible input set be compact. The appropriate notion to investigate in this framework is seen to be that of approximate robustness. Roughly speaking, an approximately robust system is one for which the output can be guaranteed to lie "[var epsilon]-close" to a prespecified set at some future time T > 0. This guarantee must hold for all admissible (possibly time-varying) variations in the values of the uncertain parameters. The principal result of this paper is a necessary and sufficient condition for approximate robustness. To "test" this condition, one must solve a finite-dimensional optimization problem over a compact domain, the unit simplex. Such a result is tantamount to a major reduction in the complexity of the problem; i.e., the original robustness problem which is infinite-dimensional admits a finite-dimensional parameterization. It is also shown how this theory specializes to the existing theory of Barmish and Barmish and Lin under the imposition of additional assumptions. A number of illustrative examples and special cases are presented. A detailed computer implementation of the theory is also discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24205/1/0000464.pd

On solving generalized convex MINLP problems using supporting hyperplane techniques

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke’s subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable f ∘  f∘ -quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an f ∘  f∘ -pseudoconvex function, subject to f ∘  f∘ -pseudoconvex constraints. With some additional conditions, the proof is also valid for f ∘  f∘ -quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443–459, 2017) to also solve problems with f ∘  f∘ -pseudoconvex objective function.</p

Using projected cutting planes in the extended cutting plane method

In this paper we show that simple projections can improve the algorithmic performance of cutting plane-based optimization methods. Projected cutting planes can, for example, be used as alternatives to standard cutting planes or supporting hyperplanes in the extended cutting plane (ECP) method. In the paper we analyse the properties of such an algorithm and prove that it will converge to a global optimum for smooth and nonsmooth convex mixed integer nonlinear programming problems. Additionally, we show that we are able to solve two old but very difficult facility layout problems (FLP), with previously unknown optimal solutions, to verified global optimum by using projected cutting planes in the algorithm. These solution results are also given in the paper

Global Optimisation for Energy System

The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity