Traffic assignment: on the interplay between optimization and equilibrium problems

Population-based approaches have given us new search strategies and ideas in order to solve optimization problems. Usually, these methods are based on the performance carried out by a finite number of agents, which by the interaction between them they evolve and work all over the search space. Also, it is well-known that the correct employment of parameter values in this kind of method can positively impact their performance and behavior. In this context, the present work focuses on the design of a hybrid architecture which smartly balances the population size on run-time. In order to smartly balance and control the population size, a modular approach, named Linear Modular Population Balancer (LMPB), is proposed. The main ideas behind the designed architecture include the solving strategy behind a population-based metaheuristic, the influence of learning components based on multiple statistical modeling methods which transform the dynamic data generated into knowledge, and the possibilities to tackle both discrete and continuous optimization problems. In this regard, three modules are proposed for LMPB, which concern tasks such as the management of the population-based algorithm, parameter setting, probabilities, learning methods, and selection mechanism for the population size to employ. In order to test the viability and effectiveness of our proposed approach, we solve a set of well-known benchmark functions and the multidimensional knapsack problem (MKP). Additionally, we illustrate promising solving results, compare them against state-of-the-art methods which have proved to be good options for solving optimization problems, and give solid arguments for future work in the necessity to keep evolving this type of proposed architecture.

Motorists often face the dilemma of choosing the route enabling them to realise the fastest (i.e., shortest) journey time. In this paper we examine discrete and continuous optimisation and equilibrium-type problems for a simplified parallel link traffic model using a variance based approach. Various methodologies used for solving these problems (brute force, dynamic programming, tabu search, steepest descent) are explored and comparison is made with the Beckmann cost function employed in transport modelling.

Large scale optimization strategies have evolved considerably over the past two decades. Currently, continuous variable optimization problems (nonlinear programs) are solved on-line for steady state refinery models with several hundred thousand variables. Moreover, efficient NLP strategies have been developed for dynamic optimization problems. Still, to take the next step, on-line optimization of large dynamic chemical processes, a number of limitations and research challenges must be overcome. Many of the advances in NLP algorithms have taken place by recognizing and exploiting the framework of Successive Quadratic Programming (SQP) algorithms. These are extensions of Newton type methods for converging to the solution of the KKT (optimality) conditions of the optimization problem. Because of this, fast convergence can be expected and a number of standard devices can be added to stabilize the algorithm to converge from poor starting points. Limitations of these Newton-based methods are also well-known. They experience difficulties in the presence of ill conditioning and extreme nonlinearities. Also, for optimization algorithms, nonconvexity can also lead to a number of difficulties and there is a need for software that allows exploitable structures for specific problem classes. A number of innovations in algorithm design and problem formulation address these issues and greatly improve performance. As a result, very fast NLP algorithms can be derived for data reconciliation, parameter estimation, nonlinear model predictive control and dynamic optimization. Moreover, inequality constraints and variable bounds can be treated through advances in interior point strategies. These methods preserve the particular problem structure and scale well in performance for large-scale problems with many constraints. Finally, the ability to solve nonlinear programs quickly also allows us to consider more challenging problems. These include the extension to solving nonconvex NLPs globally and the ability to assess the solution’s tolerance to uncertainty by considering features of nonlinear programming sensitivity analysis and robust optimization through flexibility analysis and design.

Model-based methods for continuous and discrete global optimization

In this paper we present two recent metaheuristics, particle swarm optimization and differential evolution algorithms, to solve the single machine total weighted tardiness problem, which is a typical discrete combinatorial optimization problem. Most of the literature on both algorithms is concerned with continuous optimization problems, while a few deal with discrete combinatorial optimization problems. A heuristic rule, the smallest position value (SPV) rule, borrowed from the random key representation in genetic algorithms, is developed to enable the continuous particle swarm optimization and differential evolution algorithms to be applied to all permutation types of discrete combinatorial optimization problems. The performance of these two recent population based algorithms is evaluated on widely used benchmarks from the OR library. The computational results show that both algorithms show promise in solving permutation problems. In addition, a simple but very efficient local search method based on the variable neighbourhood search (VNS) is embedded in both algorithms to improve the solution quality and the computational efficiency. Ultimately, all the best known or optimal solutions of instances are found by the VNS version of both algorithms.

Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

Memetic algorithms arise as very effective algorithms to obtain reliable and high accurate solutions for complex continuous optimization problems. Nowadays, high dimensional optimization problems are an interesting field of research. The high dimensionality introduces new problems for the optimization process, making recommendable to test the behavior of the optimization algorithms to large-scale problems. The local search method must be applied with a higher intensity, specially to most promising solutions, to explore the higher domain space around each solution. In this work, we present a preliminar study of a memetic algorithm that assigns to each individual a local search intensity that depends on its features, by chaining different local search applications. This algorithm have obtained good results in continuous optimization and we study whether is a good algorithm for large scale optimizations problems. We make experiments of our proposal using the benchmark problems defined in the Special Session or Competition on Large Scale Global Optimisation, on the IEEE Congress on Evolutionary Computation in 2008. First, we test different local search methods to identify the best one. Then, we compare the proposed algorithm with the algorithms used into the competition, obtaining that our proposal is a very promising algorithm for this type of high-dimensional problems: with dimension 500 our proposal is the second best of the compared algorithms, and the best memetic algorithm.

Nash equilibria in discrete routing games with convex latency functions

Optimization problems with continuous variables are very often in real-life scenarios. This is why several studies have focused on the use of metaheuristics to solve these problems, known as continuous optimization problems. This chapter focuses on the robust population-based metaheuristic Variable Mesh Optimization (VMO). We review the fundamental aspects of VMO for continuous optimization. Moreover, we analyze the variants of VMO that have been proposed to solve multimodal problems, i.e. those with multiple optima at several search subspaces. The chapter uses a homogeneous algorithmic framework to describe VMO and its variants with niching strategies for multimodal problems.KeywordsMetaheuristicsVariable mesh optimizationContinuous optimization problemsMultimodal optimization problemsNiching

The numerical solution of optimization problems with partial differential equation (PDE) constraints is vital to a growing number of science and engineering applications. The development of robust and efficient algorithms for the solution of these optimization problems presents many challenges that arise out of, e.g., the intricate mathematical structure of these problems, the complicated interactions between numerical methods for PDE and optimization, the large-scale of the optimization problems, and the increasing complexity of applications. To identify and overcome these challenges an integrated approach is needed that builds on a variety of mathematical sub-disciplines, such as theory of PDEs, distributed parameter systems, numerical solution of PDEs, numerical optimization, and numerical linear algebra. This international workshop has brought together some of the leading experts in the fast developing field of optimization problems with PDE constraints to present recent developments in this area as well as to identify open problems and further research needs. Among the themes of this workshop were the design and analysis of approaches for the solution of PDE constrained optimization problems with additional point-wise constraints on controls and states (the solution of the governing PDE). State constrained problems are particularly challenging because of the low regularity properties of the Lagrange multipliers associated with point-wise constraints on the states. A second theme was the development of adaptive methods for the solution of PDE constrained optimization problems and, more generally, the development of optimization level model reduction techniques for these problems. The goal here is to develop models (through, e.g., mesh adaptation or proper orthogonal model reduction) of the PDE constrained optimization problems that capture the relevant features of the optimization problems with a specified accuracy, but involve as few degrees of freedom as possible and, hence, are computationally less expensive to work with. A third theme was The efficient solution of linear systems arising in optimization algorithms for discretized PDE constrained optimization problems represented another theme. Finally, a number of talks presented advances and challenges in the solution of PDE constrained optimization problems arising in important science and industrial applications.

ABSTRACTWater cycle algorithm (WCA) is a population-based metaheuristic algorithm, inspired by the water cycle process and movement of rivers and streams towards sea. The WCA shows good performance in both exploration and exploitation phases. Further, the relationship between improvised exploitation and each parameter under asymmetric interval is derived and an iterative convergence of WCA is proved theoretically. In this paper, CEC’15 computationally expensive benchmark problems (i.e., 15 problems) have been considered for efficiency measurement of WCA accompanied with other optimisers. Also, a new discretisation strategy for the WCA has been proposed and applied along with other optimisers for solving combinatorial Internet shopping optimisation problem. By applying complexity analysis, it shows that using the WCA intricacy from dimension 10–30 is increased for almost three times. Proposing a unique discretisation approach along with providing iterative convergence proof can be considered as novelty of this research. By observing the attained numerical results, the WCA could find the minimum average error of CEC’15 in 12 and 8 out of 15 cases for dimensions 10 and 30, respectively. Experimental optimisation results for a wide range computationally expensive problems reveal the effectiveness and advantage of WCA for solving both continuous and discrete optimisation problems.

We investigate the worst case delay ratio between the Nash equilibrium and the social optimum in networks of N parallel links (routes) with unbounded delay functions. We compute this ratio, known as the "price of anarchy", for the case when the link delay functions correspond to M/M/ 1-FCFS or M/G/ 1-PS. For this problem, we find that the price of anarchy depends on the network topology in the sense that it is precisely equal to N. We then extend our results to M/G/ 1-FCFS and G/G/ 1-FCFS delay functions and compute the price of anarchy in a heavy load regime. Our results indicate that, even in very simple topological settings, the price of selfish behavior can potentially be very high.

The constrained convex minimization problem is to find a pointx∗with the property thatx∗∈C, andh(x∗)=min h(x),∀x∈C, whereCis a nonempty, closed, and convex subset of a real Hilbert spaceH,h(x)is a real-valued convex function, andh(x)is not Fréchet differentiable, but lower semicontinuous. In this paper, we discuss an iterative algorithm which is different from traditional gradient-projection algorithms. We firstly construct a bifunctionF1(x,y)defined asF1(x,y)=h(y)−h(x). And we ensure the equilibrium problem forF1(x,y)equivalent to the above optimization problem. Then we use iterative methods for equilibrium problems to study the above optimization problem. Based on Jung’s method (2011), we propose a general approximation method and prove the strong convergence of our algorithm to a solution of the above optimization problem. In addition, we apply the proposed iterative algorithm for finding a solution of the split feasibility problem and establish the strong convergence theorem. The results of this paper extend and improve some existing results.

Optimization of formworks shoring location as a continuous optimization problem

We explore a general-purpose heuristic algorithm for finding high-quality solutions to continuous optimization problems. The method, called continuous extremal optimization (CEO), can be considered as an extension of extremal optimization and consists of two components, one which is responsible for global searching and the other which is responsible for local searching. The CEO's performance proves competitive with some more elaborate stochastic optimization procedures such as simulated annealing, genetic algorithms, and so on. We demonstrate it on a well-known continuous optimization problem: the Lennard-Jones cluster optimization problem.