Discrete-continuous Optimization Research Articles

Selection of optimal feed trays in a closed-loop system of distillation columns

A new approach is suggested to solve the problem of discrete-continuous optimization of a distillation column system involving both continuous regime search variables and discrete variables (feed tray numbers). The approach is based on the simultaneous use of the branch-and-bound algorithm and the structural parameter method. The efficiency of this approach is demonstrated by solving the problem of discrete-continuous optimization of a system for separating the ethylene-propane fraction in ethylene production.

Design optimization of hybrid material structures

Design optimization of hybrid material components and structures often leads to multi-criteria and mixed continuous-discrete optimization problems. Moreover, for achieving readily implementable design solutions, it is highly important to consider interactions in structural and material behavior (e.g., thermo-elastic mismatches) together with thermo-mechanical and manufacturing aspects simultaneously. For the latter, response surface approximations are utilized which are also based on quantification of qualitative knowledge via fuzzy set rules. From a series of practical applications solved by Evolutionary Strategy Algorithms general conclusions for hybrid material component optimization and design are derived.

Discrete-Continuous Configuration Optimization Methods for Structures Using the Harmony Search Algorithm

This paper proposes an efficient structural optimization methods based on the harmony search (HS) heuristic algorithm that treat integrated discrete sizing and continuous geometric variables. The recently developed HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so derivative information is unnecessary. A benchmark truss example is presented to demonstrate the effectiveness and robustness of the new method, as compared to current optimization methods. The numerical results reveal that the proposed method is a powerful search and design optimization technique for structures with discrete member sizes, and may yield better solutions than those obtained using current methods.

Rendezvous phasing special-point maneuvers mixed discrete-continuous optimization using simulated annealing

The design of a rendezvous phasing special-point maneuvers can be formulated as a mixed one-zero, integer and continuous design variables problem. A hybrid approach combining simulated annealing (SA) with Newton's method is proposed for solving this problem. An integer-coded SA is used to handle the discrete design variables, while Newton's method is applied to handle the continuous design variables. Three improvements are employed to make the proposed approach more efficient and robust. A two-day low-earth rendezvous-phasing problem is used as an example. The effects of the improvements are investigated. The results show that the SA based hybrid approach is very effective and efficient and the SA has better convergence ability compared with an integer-coded genetic algorithm.

Continuous reformulations of discrete–continuous optimization problems

This paper treats the solution of nonlinear optimization problems involving discrete decision variables, also known as generalized disjunctive programming (GDP) or mixed-integer nonlinear programming (MINLP) problems, that arise in process engineering. The key idea is to eliminate the discrete decision variables by adding a set of continuous variables and constraints that represent the discrete decision space of the optimization problem. With such a reformulation, we are able to apply solution algorithms for purely continuous nonlinear optimization problems to efficiently calculate local minima of GDP or MINLP problems. In this contribution, we propose different alternatives to reformulate GDP/MINLP problems as continuous optimization problems. We furthermore investigate theoretical properties of the different reformulations with regard to their numerical solution. The proposed formulations are illustrated and analyzed on the basis of optimization problems dealing with process engineering applications involving stationary as well as dynamic process models.

Discrete-continuous structural optimization with distributed evolution strategies

This paper describes the development and application of distributed Evolution Strategies (ES) in the field of mixed-discrete structural optimization. ES are direct, probabilistic optimization methods based on the use of the three evolution operators recombination, mutation and selection in an evolving population of competing individuals in the design space. Advanced features of the employed ES include self-adaption of strategy parameters to achieve on-line tuning of the optimization process and a flexible selection scheme that allows scalable lifetimes of individuals The property of using a population of coexisting but independent individuals allows the efficient realization in distributed computing environments. This approach results in a drastic reduction of response time and an improved applicability for large scale problems

AN IMPROVED GENETIC ALGORITHM FOR CONTINUOUS AND MIXED DISCRETE-CONTINUOUS OPTIMIZATION

Modifications to the standard genetic algorithm through a finetuning strategy, a hillclimbing strategy and the use of independent subpopulations coupled with shuffling are described. The improvements obtained are demonstrated using two optimization problems; a continuous variable rainfall-runoff model calibration and a previously-studied mixed discrete-continuous optimization for cost minimization in pressure vessel manufacture. The use of independent Subpopulations and shuffling is found to considerably improve optimizations of the two problems whilst the finetuning and hillclimbing notably improve optimization in the model calibration but not the pressure vessel cost minimization. In the rainfall-runoff modelling the parameter sets obtained by the improved genetic algorithm are more consistent and seem more informative than those obtained with the standard genetic algorithm. With the pressure vessel design problem, lower costs are obtained than in previous studies.

Vertex finding with deformable templates at LHC

We present a novel vertex finding technique. The task is formulated as a discrete-continuous optimisation problem in a way similar to the deformable templates approach for the track finding. Unlike the track finding problem, “elastic hedgehogs” rather than elastic arms are used as deformable templates. They are initialised by a set of procedures which provide zero level approximation for vertex positions and track parameters at the vertex point. The algorithm was evaluated using the simulated events for the LHC CMS detector and demonstrated good performance.

An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and Its Applications to Mechanical Design

An algorithm for solving nonlinear optimization problems involving discrete, integer, zero-one, and continuous variables is presented. The augmented Lagrange multiplier method combined with Powell’s method and Fletcher and Reeves Conjugate Gradient method are used to solve the optimization problem where penalties are imposed on the constraints for integer/discrete violations. The use of zero-one variables as a tool for conceptual design optimization is also described with an example. Several case studies have been presented to illustrate the practical use of this algorithm. The results obtained are compared with those obtained by the Branch and Bound algorithm. Also, a comparison is made between the use of Powell’s method (zeroth order) and the Conjugate Gradient method (first order) in the solution of these mixed variable optimization problems.

APROS: Algorithmic Development Methodology for Discrete-Continuous Optimization Problems

APROS represents an algorithmic development procedure for classes of mathematical programming problems that involve some form of decomposition technique and require extensive communication of data between a set of subproblems whose sizes and structures may vary during the solution procedure. Examples include most classes of mixed-integer nonlinear programming problems and large-scale mixed-integer linear programming problems as well as a wide variety of algorithms for large-scale nonlinear and linear programming problems exhibiting special structure. APROS works through the General Algebraic Modeling System (GAMS) to provide exact syntactic statement of algorithmic solution procedures. APROS procedures are implemented through GAMS which is interfaced with nonlinear, linear and mixed-integer linear programming solvers to provide completely general automated implementations of many well known algorithms including the Generalized Benders Decomposition, the Outer Approximation/Equality Relaxation and Dantzig-Wolfe Decomposition. The flexibility of APROS and the highly procedural modeling language GAMS provide a means of obtaining quick solutions to difficult classes of problems by implementing a selected algorithm, and also provide a unique tool for developing, prototyping and experimenting with new algorithms. A description of the procedural components, features and setups is presented with emphasis on the generality of the techniques.

Enterprise Profit Maximization in the Range of Products and Production Rate Selection Problems

In conditions of monopolistic position of enterprises, they exhibit unrestrained tendency to increase the prices of their products, without taking care for full utilization of their production capacities. Such a situation may occur in the planned and as well as the free-market national economy. The decision center is able to counteract this tendency by introducing a proper system of income taxes, which results in the optimal prices providing the maximal profit. Simultaneously, due to the high level of the prime costs of small production series, there exists some threshold value in the relation between profit and production rate. If the production rate is less than this value, it is unprofitable. For reasons mentioned, the problem of enterprise profit optimization in the case of production capacities constrained, is a non-trivial mathematical programming problem. It is a mixed discrete-continuous optimization problem: the decision variables connected with selecting ranges of products are of discrete (zero-one) type: the decision variables associated with the choice of the optimal production rate are continuous. The paper presents an optimization algorithm which can be used to solve this mixed discrete-continuous decision problem. The worked out method can be also applied to other socio-economic decision problems of similar type.