Study of hybrid strategies for multi-objective optimization using gradient based methods and evolutionary algorithms

Most of the optimization problems encountered in engineering have conflicting objectives. In order to solve these problems, genetic algorithms (GAs) and gradient-based methods are widely used. GAs are relatively easy to implement, because these algorithms only require first-order information of the objectives and constraints. On the other hand, GAs do not have a standard termination condition and therefore they may not converge to the exact solutions. Gradient-based methods, on the other hand, are based on first- and higher-order information of the objectives and constraints. These algorithms converge faster to the exact solutions in solving single-objective optimization problems, but are inefficient for multi-objective optimization problems (MOOPs) and unable to solve those with non-convex objective spaces. The work in this dissertation focuses on developing a hybrid strategy for solving MOOPs based on feasible sequential quadratic programming (FSQP) and nondominated sorting genetic algorithm II (NSGA-II). The hybrid algorithms developed in this dissertation are tested using benchmark problems and evaluated based on solution distribution, solution accuracy, and execution time. Based on these performance factors, the best hybrid strategy is determined and found to be generally efficient with good solution distributions in most of the cases studied. The best hybrid algorithm is applied to the design of a crushing tube and is shown to have relatively well-distributed solutions and good efficiency compared to solutions obtained by NSGA-II and FSQP alone

Discovering Valuable Items from Massive Data

Suppose there is a large collection of items, each with an associated cost and an inherent utility that is revealed only once we commit to selecting it. Given a budget on the cumulative cost of the selected items, how can we pick a subset of maximal value? This task generalizes several important problems such as multi-arm bandits, active search and the knapsack problem. We present an algorithm, GP-Select, which utilizes prior knowledge about similarity be- tween items, expressed as a kernel function. GP-Select uses Gaussian process prediction to balance exploration (estimating the unknown value of items) and exploitation (selecting items of high value). We extend GP-Select to be able to discover sets that simultaneously have high utility and are diverse. Our preference for diversity can be specified as an arbitrary monotone submodular function that quantifies the diminishing returns obtained when selecting similar items. Furthermore, we exploit the structure of the model updates to achieve an order of magnitude (up to 40X) speedup in our experiments without resorting to approximations. We provide strong guarantees on the performance of GP-Select and apply it to three real-world case studies of industrial relevance: (1) Refreshing a repository of prices in a Global Distribution System for the travel industry, (2) Identifying diverse, binding-affine peptides in a vaccine de- sign task and (3) Maximizing clicks in a web-scale recommender system by recommending items to users

The Grand Experiment of Communism: Discovering the Trade-off between Equality and Efficiency

This paper aims to explain the rise and fall of communism by exploring the interplay between economic incentives and social preferences in different economic systems. We introduce inequality-averse and inefficiency-averse agents responding to economic incentives and transmitting their ideology as they are affected by evolving outcomes. We analyze their conflict through the interaction between leaders with economic power and followers with ideological determination. The socioeconomic dynamics of our model generate a pendulum-like switch from markets to a centrally-planned economy abolishing private ownership, and back to restoring market incentives. The grand experiment of communism is thus characterized to have led to the discovery of a trade-off between equality and efficiency at the scale of alternative economic systems.capitalism, communism, inequality, inefficiency, ideological transmission, economic transititions

Association of Christians in the Mathematical Sciences Proceedings 2019

The conference proceedings of the Association of Christians in the Mathematical Sciences biannual conference, May 29-June 1, 2019 at Indiana Wesleyan University

The Grand Experiment of Communism: Discovering the Trade-off between Equality and Efficiency

This paper aims to explain the rise and fall of communism by exploring the interplay between economic incentives and social preferences transmitted by ideology. We introduce inequality-averse and inefficiency-averse agents and analyze their conflict through the interaction between leaders with economic power and followers with ideological determination. The socioeconomic dynamics of our model generate a pendulum-like switch from markets to a centrally-planned economy abolishing private ownership, and back to restoring market incentives. The grand experiment of communism is thus characterized to have led to the discovery of a trade-off between equality and efficiency at the scale of alternative economic systems. While our focus is on the long-run transitions from capitalism to communism and back observed in the course of the 20-th century, the model also derives conditions under which the two systems converge and become stable.

Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability

The purpose of this thesis is twofold. The first and major part is devoted to sensitivity analysis of various discrete optimization problems while the second part addresses methods applied for calculating measures of solution stability and solving multicriteria discrete optimization problems. Despite numerous approaches to stability analysis of discrete optimization problems two major directions can be single out: quantitative and qualitative. Qualitative sensitivity analysis is conducted for multicriteria discrete optimization problems with minisum, minimax and minimin partial criteria. The main results obtained here are necessary and sufficient conditions for different stability types of optimal solutions (or a set of optimal solutions) of the considered problems. Within the framework of quantitative direction various measures of solution stability are investigated. A formula for a quantitative characteristic called stability radius is obtained for the generalized equilibrium situation invariant to changes of game parameters in the case of the H¨older metric. Quality of the problem solution can also be described in terms of robustness analysis. In this work the concepts of accuracy and robustness tolerances are presented for a strategic game with a finite number of players where initial coefficients (costs) of linear payoff functions are subject to perturbations. Investigation of stability radius also aims to devise methods for its calculation. A new metaheuristic approach is derived for calculation of stability radius of an optimal solution to the shortest path problem. The main advantage of the developed method is that it can be potentially applicable for calculating stability radii of NP-hard problems. The last chapter of the thesis focuses on deriving innovative methods based on interactive optimization approach for solving multicriteria combinatorial optimization problems. The key idea of the proposed approach is to utilize a parameterized achievement scalarizing function for solution calculation and to direct interactive procedure by changing weighting coefficients of this function. In order to illustrate the introduced ideas a decision making process is simulated for three objective median location problem. The concepts, models, and ideas collected and analyzed in this thesis create a good and relevant grounds for developing more complicated and integrated models of postoptimal analysis and solving the most computationally challenging problems related to it.Siirretty Doriast

Comparing Income Distributions Between Economies That Reward Innovation And Those That Reward Knowledge

In this paper, we develop an optimal control model of labor allocation in two types of economy - one economy is for innovative workers and the other one for knowledge workers. In both economies, workers allocate time between learning and discovering new knowledge. Both markets consist of a continuum of heterogeneous agents that are distinguished by their learning ability. Workers are rewarded for the knowledge they possess in the knowledge economy, and only for the new knowledge they create in the innovative economy. We show that, at steady state, while human capital accumulation is higher in the knowledge economy, the rate of knowledge creation is not necessarily higher in the innovative economy. Secondly, we prove that when the cost of learning is sufficiently high, the distribution of net wage income in the knowledge economy dominates that in the innovative economy in the first degree.

A review of portfolio planning: Models and systems

In this chapter, we first provide an overview of a number of portfolio planning models which have been proposed and investigated over the last forty years. We revisit the mean-variance (M-V) model of Markowitz and the construction of the risk-return efficient frontier. A piecewise linear approximation of the problem through a reformulation involving diagonalisation of the quadratic form into a variable separable function is also considered. A few other models, such as, the Mean Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the Minimax (MM) model which use alternative metrics for risk are also introduced, compared and contrasted. Recently asymmetric measures of risk have gained in importance; we consider a generic representation and a number of alternative symmetric and asymmetric measures of risk which find use in the evaluation of portfolios. There are a number of modelling and computational considerations which have been introduced into practical portfolio planning problems. These include: (a) buy-in thresholds for assets, (b) restriction on the number of assets (cardinality constraints), (c) transaction roundlot restrictions. Practical portfolio models may also include (d) dedication of cashflow streams, and, (e) immunization which involves duration matching and convexity constraints. The modelling issues in respect of these features are discussed. Many of these features lead to discrete restrictions involving zero-one and general integer variables which make the resulting model a quadratic mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the algorithms and solution methods for this class of problems are also discussed. The issues of preparing the analytic data (financial datamarts) for this family of portfolio planning problems are examined. We finally present computational results which provide some indication of the state-of-the-art in the solution of portfolio optimisation problems