Classification-Based Method of Linear Multicriteria Optimization

The paper describes a classification-based learning-oriented interactive method for solving linear multicriteria optimization problems. The method allows the decision makers describe their preferences with greater flexibility, accuracy and reliability. The method is realized in an experimental software system supporting the solution of multicriteria optimization problems.

l-stable Functions and Constrained Optimization

Иван Гинчев - Класът на ℓ-устойчивите в точка функции, дефиниран в [2] и разширяващ класа на C1,1 функциите, се обобщава от скаларни за векторни функции. Доказани са някои свойства на ℓ-устойчивите векторни функции. Показано е, че векторни оптимизационни задачи с ограничения допускат условия от втори ред изразени чрез посочни производни, което обобщава резултати от [2] и [5].

On Pareto Sets in Multi-Criteria Optimization

Здравко Д. Славов - В тази работа се разглеждат Паретовските решения в непрекъсната многокритериална оптимизация. Обсъжда се ролята на някои предположения, които влияят на характеристиките на Паретовските множества. Авторът се е опитал да премахне предположенията за вдлъбнатост на целевите функции и изпъкналост на допустимата област, които обикновено се използват в многокритериалната оптимизация. Резултатите са на базата на конструирането на ретракция от допустимата област върху Парето-оптималното множество.

Hybrid multi-objective optimization and hybridized self-organizing response surface method

Numerical optimization is a technique where a computer is used to explore design parameter combinations to find extremes in performance factors. In multi-objective optimization several performance factors can be optimized simultaneously. The solution to multi-objective optimization problems is not a single design, but a family of optimized designs referred to as the Pareto frontier. The Pareto frontier is a trade-off curve in the objective function space composed of solutions where performance in one objective function is traded for performance in others. A Multi-Objective Hybridized Optimizer (MOHO) was created for the purpose of solving multi-objective optimization problems by utilizing a set of constituent optimization algorithms. MOHO tracks the progress of the Pareto frontier approximation development and automatically switches amongst those constituent evolutionary optimization algorithms to speed the formation of an accurate Pareto frontier approximation. Aerodynamic shape optimization is one of the oldest applications of numerical optimization. MOHO was used to perform shape optimization on a 0.5-inch ballistic penetrator traveling at Mach number 2.5. Two objectives were simultaneously optimized: minimize aerodynamic drag and maximize penetrator volume. This problem was solved twice. The first time the problem was solved by using Modified Newton Impact Theory (MNIT) to determine the pressure drag on the penetrator. In the second solution, a Parabolized Navier-Stokes (PNS) solver that includes viscosity was used to evaluate the drag on the penetrator. The studies show the difference in the optimized penetrator shapes when viscosity is absent and present in the optimization. In modern optimization problems, objective function evaluations may require many hours on a computer cluster to perform these types of analysis. One solution is to create a response surface that models the behavior of the objective function. Once enough data about the behavior of the objective function has been collected, a response surface can be used to represent the actual objective function in the optimization process. The Hybrid Self-Organizing Response Surface Method (HYBSORSM) algorithm was developed and used to make response surfaces of objective functions. HYBSORSM was evaluated using a suite of 295 non-linear functions. These functions involve from 2 to 100 variables demonstrating robustness and accuracy of HYBSORSM.

Modified continuous ant colony algorithm for function optimization

Many classical as well as modern optimization techniques exist. One such modern method belonging to the field of swarm intelligence is termed ant colony optimization. This relatively new concept in optimization involves the use of artificial ants and is based on real ant behavior inspired by the way ants search for food. In this thesis, a novel ant colony optimization technique for continuous domains was developed. The goal was to provide improvements in computing time and robustness when compared to other optimization algorithms. Optimization function spaces can have extreme topologies and are therefore difficult to optimize. The proposed method effectively searched the domain and solved difficult single-objective optimization problems. The developed algorithm was run for numerous classic test cases for both single and multi-objective problems. The results demonstrate that the method is robust, stable, and that the number of objective function evaluations is comparable to other optimization algorithms.

Robust scenario formulations for strategic supply chain optimization under uncertainty

Strategic supply chain optimization (SCO) problems are often modelled as a two-stage optimization problem, in which the first-stage variables represent decisions on the development of the supply chain and the second-stage variables represent decisions on the operations of the supply chain. When uncertainty is explicitly considered, the problem becomes an intractable infinite-dimensional optimization problem, which is usually solved approximately via a scenario or a robust approach. This paper proposes a novel synergy of the scenario and robust approaches for strategic SCO under uncertainty. Two formulations are developed, namely, naïve robust scenario formulation and affinely adjustable robust scenario formulation. It is shown that both formulations can be reformulated into tractable deterministic optimization problems if the uncertainty is bounded with the infinity-norm, and the uncertain equality constraints can be reformulated into deterministic constraints without assumption of the uncertainty region. Case studies of a classical farm planning problem and an energy and bioproduct SCO problem demonstrate the advantages of the proposed formulations over the classical scenario formulation. The proposed formulations not only can generate solutions with guaranteed feasibility or indicate infeasibility of a problem, but also can achieve optimal expected economic performance with smaller numbers of scenarios.

Differential Evolution: A Tool for Global Optimization

This report describes a tool for global optimization that implements the Differential Evolution optimization algorithm as a new Excel add-in. The tool takes a step beyond Excel’s Solver add-in, because Solver often returns a local minimum, that is, a minimum that is less than or equal to nearby points, while Differential Evolution solves for the global minimum, which includes all feasible points. Despite complex underlying mathematics, the tool is relatively easy to use, and can be applied to practical optimization problems, such as establishing pricing and awards in a hotel loyalty program. The report demonstrates an example of how to develop an optimum approach to that problem.

Problems of optimal transportation on the circle and their mechanical applications

We consider a mechanical problem concerning a 2D axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry. The body moves through a medium of non-interacting particles at rest, and collisions of particles with the body's boundary are perfectly elastic (billiard-like). The body has a blunt nose: a line segment orthogonal to the symmetry axis. It is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance. This problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transfer problem on the circle with the transportation cost being the squared Euclidean distance. We find the exact solution for this problem when the amplitude of rotation is smaller than a fixed critical value, and give a numerical solution otherwise. As a by-product, we get explicit description of the solution for a class of optimal transfer problems on the circle.

Combination of Evolutionary Algorithms with Experimental Design, Traditional Optimization and Machine Learning

Evolutionary algorithms alone cannot solve optimization problems very efficiently since there are many random (not very rational) decisions in these algorithms. Combination of evolutionary algorithms and other techniques have been proven to be an efficient optimization methodology. In this talk, I will explain the basic ideas of our three algorithms along this line (1): Orthogonal genetic algorithm which treats crossover/mutation as an experimental design problem, (2) Multiobjective evolutionary algorithm based on decomposition (MOEA/D) which uses decomposition techniques from traditional mathematical programming in multiobjective optimization evolutionary algorithm, and (3) Regular model based multiobjective estimation of distribution algorithms (RM-MEDA) which uses the regular property and machine learning methods for improving multiobjective evolutionary algorithms.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Dynamic Multi-Objective Optimization With jMetal and Spark: a Case Study

Technologies for Big Data and Data Science are receiving increasing research interest nowadays. This paper introduces the prototyping architecture of a tool aimed to solve Big Data Optimization problems. Our tool combines the jMetal framework for multi-objective optimization with Apache Spark, a technology that is gaining momentum. In particular, we make use of the streaming facilities of Spark to feed an optimization problem with data from different sources. We demonstrate the use of our tool by solving a dynamic bi-objective instance of the Traveling Salesman Problem (TSP) based on near real-time traffic data from New York City, which is updated several times per minute. Our experiment shows that both jMetal and Spark can be integrated providing a software platform to deal with dynamic multi-optimization problems.

On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

The Kidney Exchange Problem (KEP) is a combinatorial optimization problem and has attracted the attention from the community of integer programming/combinatorial optimisation in the past few years. Defined on a directed graph, the KEP has two variations: one concerns cycles only, and the other, cycles as well as chains on the same graph. We call the former a Cardinality Constrained Multi-cycle Problem (CCMcP) and the latter a Cardinality Constrained Cycles and Chains Problem (CCCCP). The cardinality for cycles is restricted in both CCMcP and CCCCP. As for chains, some studies in the literature considered cardinality restrictions, whereas others did not. The CCMcP can be viewed as an Asymmetric Travelling Salesman Problem that does allow subtours, however these subtours are constrained by cardinality, and that it is not necessary to visit all vertices. In existing literature of the KEP, the cardinality constraint for cycles is usually considered to be small (to the best of our knowledge, no more than six). In a CCCCP, each vertex on the directed graph can be included in at most one cycle or chain, but not both. The CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights, particularly due to their similarities to some travelling salesman- and vehicle routing-family of problems. In this paper, our main focus is to review the existing mathematical programming models and solution methods in the literature, analyse the performance of these models, and identify future research directions. Further, we propose a polynomial-sized and an exponential-sized mixed-integer linear programming model, discuss a number of stronger constraints for cardinality-infeasible-cycle elimination for the latter, and present some preliminary numerical results.

On the proximal Landweber Newton method for a class of nonsmooth convex problems

We consider a class of nonsmooth convex optimization problems where the objective function is a convex differentiable function regularized by the sum of the group reproducing kernel norm and (Formula presented.)-norm of the problem variables. This class of problems has many applications in variable selections such as the group LASSO and sparse group LASSO. In this paper, we propose a proximal Landweber Newton method for this class of convex optimization problems, and carry out the convergence and computational complexity analysis for this method. Theoretical analysis and numerical results show that the proposed algorithm is promising.

Strategic Optimization Techniques For FRTU Deployment and Chip Physical Design

Combinatorial optimization is a complex engineering subject. Although formulation often depends on the nature of problems that differs from their setup, design, constraints, and implications, establishing a unifying framework is essential. This dissertation investigates the unique features of three important optimization problems that can span from small-scale design automation to large-scale power system planning: (1) Feeder remote terminal unit (FRTU) planning strategy by considering the cybersecurity of secondary distribution network in electrical distribution grid, (2) physical-level synthesis for microfluidic lab-on-a-chip, and (3) discrete gate sizing in very-large-scale integration (VLSI) circuit. First, an optimization technique by cross entropy is proposed to handle FRTU deployment in primary network considering cybersecurity of secondary distribution network. While it is constrained by monetary budget on the number of deployed FRTUs, the proposed algorithm identi?es pivotal locations of a distribution feeder to install the FRTUs in different time horizons. Then, multi-scale optimization techniques are proposed for digital micro?uidic lab-on-a-chip physical level synthesis. The proposed techniques handle the variation-aware lab-on-a-chip placement and routing co-design while satisfying all constraints, and considering contamination and defect. Last, the first fully polynomial time approximation scheme (FPTAS) is proposed for the delay driven discrete gate sizing problem, which explores the theoretical view since the existing works are heuristics with no performance guarantee. The intellectual contribution of the proposed methods establishes a novel paradigm bridging the gaps between professional communities.

Distributed optimization in transportation and logistics networks

Many important problems in communication networks, transportation networks, and logistics networks are solved by the minimization of cost functions. In general, these can be complex optimization problems involving many variables. However, physicists noted that in a network, a node variable (such as the amount of resources of the nodes) is connected to a set of link variables (such as the flow connecting the node), and similarly each link variable is connected to a number of (usually two) node variables. This enables one to break the problem into local components, often arriving at distributive algorithms to solve the problems. Compared with centralized algorithms, distributed algorithms have the advantages of lower computational complexity, and lower communication overhead. Since they have a faster response to local changes of the environment, they are especially useful for networks with evolving conditions. This review will cover message-passing algorithms in applications such as resource allocation, transportation networks, facility location, traffic routing, and stability of power grids.