There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Introduction to stochastic programming

In this chapter we deal with the following nonlinear fractional programming problem: 

$$P:\mathop{{\max }}\limits_{{x \in s}} q(x) = (f(x) + \alpha )/((x) + \beta )$$

where f, g: R n → R, α, β ∈ R, S ⊆ R n . To simplify things, and without restricting the generality of the problem, it is usually assumed that, g(x) + β + 0 for all x ∈ S,S is non-empty and that the objective function has a finite optimal value.

Nonlinear Fractional Programming

https://www.cirrelt.ca/DocumentsTravail/CIRRELT-2008-36.pdf

The Design of Robust Value-Creating Supply Chain Networks: A Critical Review †

Variable neighbourhood search (VNS) is a metaheuristic, or a framework for building heuristics, based upon systematic changes of neighbourhoods both in descent phase, to find a local minimum, and in perturbation phase to emerge from the corresponding valley It was first proposed in 1997 and has since then rapidly developed both in its methods and its applications In the present paper, these two aspects are thoroughly reviewed and an extensive bibliography is provided Moreover, one section is devoted to newcomers It consists of steps for developing a heuristic for any particular problem Those steps are common to the implementation of other metaheuristics

Variable neighbourhood search: methods and applications

The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way. Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems. We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values.

/pdf/an-algorithmic-framework-for-the-exact-solution-of-the-prize-4wl9z7ovbw.pdf

An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem

The simple plant location problem (SPLP) is considered and a genetic algorithm is proposed to solve this problem. By using the developed algorithm it is possible to solve SPLP with more than 1000 facility sites and customers. Computational results are presented and compared to dual based algorithms.

/pdf/solving-the-simple-plant-location-problem-by-genetic-1trttrbesp.pdf

Solving the simple plant location problem by genetic algorithm

The uncapacitated facility location (UFL) problem is one of the most famous and most studied problems in the operations research literature. Given a set of potential facility locations and a set of customers, the goal is to find a subset of facility locations to open and to allocate each customer to open facilities so that the facility opening plus customer allocation costs are minimized. In our setting, for each customer the allocation cost is assumed to be a linear or separable convex quadratic function. Motivated by recent UFL applications in business analytics, we revise approaches that work on a projected decision space and hence are intrinsically more scalable for large-scale input data. Our working hypothesis is that many of the exact (decomposition) approaches that were proposed decades ago and discarded soon after need to be redesigned to take advantage of the new hardware and software technologies. To this end, we “thin out” the classical models from the literature and use (generalized) Benders ...

/pdf/redesigning-benders-decomposition-for-large-scale-facility-vm16ttm863.pdf

Redesigning Benders Decomposition for Large-Scale Facility Location

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art metho...

/pdf/a-new-general-purpose-algorithm-for-mixed-integer-bilevel-mmyypxz53e.pdf

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

http://www.dei.unipd.it/~fisch/papers/Benders_for_CFL.pdf

Ivana Ljubić

Papers

An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem

Solving the simple plant location problem by genetic algorithm

Redesigning Benders Decomposition for Large-Scale Facility Location

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Benders decomposition without separability: A computational study for capacitated facility location problems