Standard Optimization Problem Research Articles

An evolutionary algorithm for multi-criteria inverse optimal value problems using a bilevel optimization model

Given a linear program, a desired optimal objective value, and a set of feasible cost vectors, one needs to determine a cost vector of the linear program such that the corresponding optimal objective value is closest to the desired value. The problem is always known as a standard inverse optimal value problem. When multiple criteria are adopted to determine cost vectors, a multi-criteria inverse optimal value problem arises, which is more general than the standard case. This paper focuses on the algorithmic approach for this class of problems, and develops an evolutionary algorithm based on a dynamic weighted aggregation method. First, the original problem is converted into a bilevel program with multiple upper level objectives, in which the lower level problem is a linear program for each fixed cost vector. In addition, the potential bases of the lower level program are encoded as chromosomes, and the weighted sum of the upper level objectives is taken as a new optimization function, by which some potential nondominated solutions can be generated. In the design of the evolutionary algorithm some specified characteristics of the problem are well utilized, such as the optimality conditions. Some preliminary computational experiments are reported, which demonstrates that the proposed algorithm is efficient and robust.

A hybrid particle swarm with a time-adaptive topology for constrained optimization

For constrained optimization problems set in a continuous space, feasible regions might be disjointed and the optimal solution might be in any of these regions. Thus, locating these feasible regions (ideally all of them) as well as identifying the most promising region (in terms of objective value) at the end of the optimization process would be of a great significance. In this paper a time-adaptive topology is proposed that enables a variant of the particle swarm optimization (PSO) to locate many feasible regions at the early stages of the optimization process and to identify the most promising one at the latter stages of the optimization process. This PSO variant is combined with two local searches which improve the ability of the algorithm in both finding feasible regions and higher quality solutions. This method is further hybridized with covariance matrix adaptation evolutionary strategy (CMA-ES) to enhance its ability to improve the solutions at the latter stages of the optimization process. Results generated by this hybrid method are compared with the results of several other state-of-the-art methods in dealing with standard benchmark constraint optimization problems.

Finding the Nearest Valid Covariance Matrix: A FX Market Case

We consider the problem of finding a valid covariance matrix in the FX market given an initial non-PSD estimate of such a matrix. The standard no-arbitrage assumption implies additional linear constraints on such matrices, which automatically makes them singular. As a result, one cannot just take the given estimate plug it into the standard optimization problem and solve it by applying even the most advanced numerical techniques developed recently. The reason is that such a problem is not well-posed while the PSD-solution is not strictly feasible. In order to deal with this issue, we described a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which turns out to be well posed and of a smaller scale. We show that after solving the reduced problem the solution to the initial problem can be uniquely recovered in one step. We run numerous numerical experiments to compare performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that virtually any numerical method can be applied effectively to find its solution.

A Proof Without Words and a Maximum Without Calculus

We define a standard optimization problem with quadratic objective function and provide a rigorous visual proof for its solution without using calculus. We then show that such standard problem is a building block for several economic models related to microeconomics, game theory and pricing strategies.

Legendre Polynomials Iterative Technique for Solving a Class of Nonlinear Optimal Control Problems

A computational algorithm is proposed to solve a class of nonlinear optimal control problems. The proposed algorithm is based on replacing the original nonlinear optimal control problem by a sequence of time-varying linear quadratic optimal control problems. This is accomplished by employing an iterative technique developed by Banks [1-5] which is based on replacing the original nonlinear system by a sequence of linear time-varying systems. Then each of the time-varying linear quadratic optimal control problems is transformed into a standard quadratic programming problem by parameterizing the state variables by a finite length Legendre polynomials with unknown parameters. The solution of a standard nonlinear optimal control problem is presented, to show the effectiveness of the proposed method.

H ∞ observer design for uncertain nonlinear discrete‐time systems with time‐delay: LMI optimization approach

SummaryWe present a robust H ∞ observer for a class of nonlinear discrete‐time systems. The class under study includes an unknown time‐varying delay limited by upper and lower bounds, as well as time‐varying parametric uncertainties. We design a nonlinear H ∞ observer, by using the upper and lower bounds of the delay, that guarantees asymptotic stability of the estimation error dynamics and is also robust against time‐varying parametric uncertainties. The described problem is converted to a standard optimization problem, which can be solved in terms of linear matrix inequalities (LMIs). Then, we expand the problem to a multi‐objective optimization problem in which the maximum admissible Lipschitz constant and the minimum disturbance attenuation level are the problem objectives. Finally, the proposed observer is illustrated with two examples. Copyright © 2014 John Wiley & Sons, Ltd.

Finding the Nearest Valid Covariance Matrix: A FX Market Case

We consider the problem of finding a valid covariance matrix in the FX market given an initial non-PSD estimate of such a matrix. The standard no-arbitrage assumption implies additional linear constraints on such matrices, which automatically makes them singular. As a result, one cannot just take the given estimate plug it into the standard optimization problem and solve it by applying even the most advanced numerical techniques developed recently. The reason is that such a problem is not well-posed while the PSD-solution is not strictly feasible. In order to deal with this issue, we described a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which turns out to be well posed and of a smaller scale. We show that after solving the reduced problem the solution to the initial problem can be uniquely recovered in one step. We run numerous numerical experiments to compare performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that virtually any numerical method can be applied effectively to find its solution.

A Numerical Method of the Euler-Bernoulli Beam with Optimal Local Kelvin-Voigt Damping

This paper deals with the numerical approximation problem of the optimal control problem governed by the Euler-Bernoulli beam equation with local Kelvin-Voigt damping, which is a nonlinear coefficient control problem with control constraints. The goal of this problem is to design a control input numerically, which is the damping and distributes locally on a subinterval of the region occupied by the beam, such that the total energy of the beam and the control on a given time period is minimal. We firstly use the finite element method (FEM) to obtain a finite-dimensional model based on the original PDE system. Then, using the control parameterization method, we approximate the finite-dimensional problem by a standard optimal parameter selection problem, which is a suboptimal problem and can be solved numerically by nonlinear mathematical programming algorithm. At last, some simulation studies will be presented by the proposed numerical approximation method in this paper, where the damping controls act on different locations of the Euler-Bernoulli beam.

Lattice Boltzmann Constraints for Standard Optimization Problems

AbstractThis paper presents a simultaneous analysis and design (SAND) approach for the control of incompressible, stationary fluid problems ruled by the lattice Boltzmann equation. Therefore the discretized lattice Boltzmann fix point problem is used as constraint for the optimization problem. This allows for the use of standard non‐linear programming techniques.To validate the proposed method a topology optimization problem is solved with the NLP solver WORHP at moderate Reynolds numbers (Re < 1000) for a 2‐dimensional flow domain. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.

A Catfish Effect Inspired Harmony Search Algorithm for Optimization

Abstract This paper describes a Catfish Effect inspired harmony search algorithm. In the algorithm, two equal size harmony memories (HM) are used to cover all the variable range, and the new harmony is generated from each history one or from the whole possible solution set, following some probability rule. After the new harmony is obtained, the worst harmony in one HM will be evaluated by the new and the best harmony in the other HM, respectively. Which means that a better harmony has the effect that causing the weak HM to better themselves. These procedures proceed on until the near-optimal solution vector is obtained. Numerical comparisons of the new algorithm with several existing harmony search algorithms are presented based on some standard benchmark optimization problems from CEC2005. The results show that the new algorithm is quite promising.

Expansion formula for fractional derivatives in variational problems

We modify the expansion formula introduced in [T.M. Atanacković, B. Stanković, An expansion formula for fractional derivatives and its applications, Fract. Calc. Appl. Anal. 7 (3) (2004) 365–378] for the left Riemann–Liouville fractional derivative in order to apply it to various problems involving fractional derivatives. As a result we obtain a new form of the fractional integration by parts formula, with the benefit of a useful approximation for the right Riemann–Liouville fractional derivative, and derive a consequence of the fractional integral inequality ∫0Ty⋅0Dtαydt≥0. Further, we use this expansion formula to transform fractional optimization (minimization of a functional involving fractional derivatives) to the standard constrained optimization problem. It is shown that when the number of terms in the approximation tends to infinity, solutions to the Euler–Lagrange equations of the transformed problem converge, in a weak sense, to solutions of the original fractional Euler–Lagrange equations. An illustrative example is treated numerically.

Joint linear processing design for distributed two-way Amplify-and-Forward MIMO relaying networks

Linear transceiver designs are investigated for distributed two-way relaying networks, which aim at minimising the Weighted Mean Square Error (WMSE) of data detections. The forwarding matrices at relays and equalization matrices at destinations are jointly optimised. To overcome the challenging limitations introduced by individual power constraints, a Semi-Definite Relaxation (SDR) called element-wise relaxation is proposed, which can transform the original optimization problem into a standard convex optimization problem. In this research, two-way relaying is understood from a pure signal processing perspective which can potentially simplify the theoretical analysis. Finally, simulation results are used for assessing the performance advantage of the proposed algorithm.

Rebalancing an investment portfolio in the presence of convex transaction costs, including market impact costs

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. We extend the standard portfolio optimization problem to consider convex transaction costs incurred when rebalancing an investment portfolio. Market impact costs measure the effect on the price of a security that result from an effort to buy or sell the security, and they can constitute a large part of the total transaction costs. The loss to a portfolio from market impact costs is often modelled with a convex function that can be expressed using second-order cone constraints. The Markowitz framework of mean-variance efficiency is used. In order to properly represent the variance of the resulting portfolio, we suggest rescaling by the funds available after paying the transaction costs. This results in a fractional programming problem, which we show can be reformulated as an equivalent convex program of size comparable to the model without transaction costs. We show that an optimal solution to the convex program can always be found that does not discard assets.

Structural Damage Identification Using an Adaptive Multi-stage Optimization Method Based on a Modified Particle Swarm Algorithm

An adaptive multi-stage optimization method utilizing a modified particle swarm optimization (MPSO) is proposed here to identify the multiple damage cases of structural systems. First the structural damage problem is defined as a standard optimization problem. An efficient objective function considering the first few natural frequencies of a structure, before and after damage, is utilized for optimization. A modified particle swarm optimization (MPSO) dealing with real values of damage variables is introduced to solve the optimization problem. In order to assess the performance of the proposed method, some illustrative examples with and without considering the measurement noise are tested. Numerical results demonstrate the high accuracy of the method proposed for determining the site and severity of multiple damage cases in the structural systems.

Simulating Glycolytic Metabolism of Escherichia coli by Using Optimal Control Models

Simulating dynamic behaviors of metabolic networks of a living cell plays an important role in system biology. As a part of dynamic model, the choice of objective function has a large influence on simulation accuracy for these behaviors. In this work, in order to precisely describe the biomass yield and substrate utilization in glycolytic metabolism of Escherichia coli, we try to modify the objective function of the existing dynamic model by using maximization of glucose utilization to replace the traditional objective one. After that, the dynamic model with the new objective is converted to a standard optimal control problem. And then we compute such model through the use of the penalty function methods. The results illustrate that the simulation curves perfectly agree with experiment data, especially with biomass concentration. Thereby, we conclude that completely utilizing substrate glucose is feasible to describe and improve the simulation accuracy on concentrations of some important metabolites in Escherichia coli. The completeness of investigating such models will be helpful and instructive for the application of bioengineering.

Planning of the Operating Points in Desalination Plants based on Energy Optimization

A methodology is developed for the optimization of the operation of Reverse Osmosis (RO) desalination process. Thus, a computer model of the process is first presented, that comprises two sub models, the first for the solution diffusion process, and the second for the effects of membrane fouling. The optimal operation problem is then stated and transformed into a mathematical optimization problem, based on the minimization of the Specific Energy Consumption (SEC): a computer-based approach is then proposed, to periodically calculate a combination of the pressure difference across the membrane, and the feed water flow rates, that minimizes the SEC, and always fulfils given operational constraints. It is shown that this mathematical problem can be transformed into a standard nonlinear optimization problem, which can be solved using off-the-self software. Application of the methodology is demonstrated on the simulation of a real RO desalination plant, which demonstrates how the proposed approach makes possible to adapt the operation to variations in the plant conditions.

A non-standard optimal control problem arising in an economics application

A recent optimal control problem in the area of economics has mathematical properties that do not fall into the standard optimal control problem formulation. In our problem the state value at the final time the state, y(T) = z, is free and unknown, and additionally the Lagrangian integrand in the functional is a piecewise constant function of the unknown value y(T). This is not a standard optimal control problem and cannot be solved using Pontryagin's Minimum Principle with the standard boundary conditions at the final time. In the standard problem a free final state y(T) yields a necessary boundary condition p(T) = 0, where p(t) is the costate. Because the integrand is a function of y(T), the new necessary condition is that y(T) should be equal to a certain integral that is a continuous function of y(T). We introduce a continuous approximation of the piecewise constant integrand function by using a hyperbolic tangent approach and solve an example using a C++ shooting algorithm with Newton iteration for solving the Two Point Boundary Value Problem (TPBVP). The minimising free value y(T) is calculated in an outer loop iteration using the Golden Section or Brent algorithm. Comparative nonlinear programming (NP) discrete-time results are also presented.

A Joint Resource Allocation Scheme for Relay Enhanced Multi-cell Orthogonal Frequency Division Multiple Networks

This paper formulates resource allocation for decode-and-forward (DF) relay assisted multi-cell orthogonal frequency division multiple (OFDM) networks as an optimization problem taking into account of inter-cell interference and users fairness. To maximize the transmit rate of system we propose a joint interference coordination, subcarrier and power allocation algorithm. To reduce the complexity, this semi-distributed algorithm divides the primal optimization into three sub-optimization problems, which transforms the mixed binary nonlinear programming problem (BNLP) into standard convex optimization problems. The first layer optimization problem is used to get the optimal subcarrier distribution index. The second is to solve the problem that how to allocate power optimally in a certain subcarrier distribution order. Based on the concept of equivalent channel gain (ECG) we transform the max-min function into standard closed expression. Subsequently, with the aid of dual decomposition, water-filling theorem and iterative power allocation algorithm the optimal solution of the original problem can be got with acceptable complexity. The third sub-problem considers dynamic co-channel interference caused by adjacent cells and redistributes resources to achieve the goal of maximizing system throughput. Finally, simulation results are provided to corroborate the proposed algorithm.

The Genetic Flock Algorithm

The purpose of this paper is to describe and evaluate a new algorithm for optimization. The new algorithm is named the Genetic Flock Algorithm. This algorithm is a type of hybrid of a Genetic Algorithm and a Particle Swarm Optimization Algorithm. The paper discusses strengths and weaknesses of these two algorithms. It then explains how the Genetic Flock Algorithm combines features of both and gives details of the algorithm. All three algorithms are compared using eight standard optimization problems that are used in the literature. It is shown that the Genetic Flock Algorithm provides superior performance on 75% of the tested cases. In the remaining 25% of the cases it outperforms either the Genetic Algorithm or the Particle Swarm Optimization Algorithm; it is never worse than both. Possible future improvements to the Genetic Flock Algorithm are briefly described.