scholarly journals Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

2021 ◽  
Author(s):  
Nathan Adelgren ◽  
Akshay Gupte
Keyword(s):  
Branch And Bound ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Splitting Method ◽  
Mixed Integer ◽  
Feasible Region ◽  
Decision Space ◽  
Conflicting Objectives ◽  
Objective Space ◽  
Single Objective

We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, checking node fathoming, presolve, and duality gap measurement. Our branch-and-bound is predominantly a decision space search method because the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and on a new set of instances that we generate using a benchmark library (MIPLIB2017) for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm. Summary of Contribution: Biobjective mixed-integer optimization problems have two linear objectives and a mixed-integer feasible region. Such problems have many applications in operations research, because many real-world optimization problems naturally comprise two conflicting objectives to optimize or can be approximated in such a manner and are even harder than single objective mixed-integer programs. Solving them exactly requires the computation of all the nondominated solutions in the objective space, whereas some applications may also require finding at least one solution in the decision space corresponding to each nondominated solution. This paper provides an exact algorithm for solving these problems using the branch-and-bound method, which works predominantly in the decision space. Of the many ingredients of this algorithm, some parts are direct extensions of the single-objective version, but the main parts are newly designed algorithms to handle the distinct challenges of optimizing over two objectives. The goal of this study is to improve solution quality and speed and show that decision-space algorithms perform comparably to, and sometimes better than, algorithms that work mainly in the objective-space.

scholarly journals Two-stage multi-tasking transform framework for large-scale many-objective optimization problems

2021 ◽  
Author(s):  
Lu Chen ◽  
Handing Wang ◽  
Wenping Ma
Keyword(s):  
Large Scale ◽  
Original Problem ◽  
Optimization Problems ◽  
Reference Points ◽  
Optimization Strategy ◽  
Two Stage ◽  
Decision Space ◽  
Second Stage ◽  
Decision Variables ◽  
Objective Space

AbstractReal-world optimization applications in complex systems always contain multiple factors to be optimized, which can be formulated as multi-objective optimization problems. These problems have been solved by many evolutionary algorithms like MOEA/D, NSGA-III, and KnEA. However, when the numbers of decision variables and objectives increase, the computation costs of those mentioned algorithms will be unaffordable. To reduce such high computation cost on large-scale many-objective optimization problems, we proposed a two-stage framework. The first stage of the proposed algorithm combines with a multi-tasking optimization strategy and a bi-directional search strategy, where the original problem is reformulated as a multi-tasking optimization problem in the decision space to enhance the convergence. To improve the diversity, in the second stage, the proposed algorithm applies multi-tasking optimization to a number of sub-problems based on reference points in the objective space. In this paper, to show the effectiveness of the proposed algorithm, we test the algorithm on the DTLZ and LSMOP problems and compare it with existing algorithms, and it outperforms other compared algorithms in most cases and shows disadvantage on both convergence and diversity.

scholarly journals A Hybrid Predictive Strategy Carried through Simultaneously from Decision Space and Objective Space for Evolutionary Dynamic Multiobjective Optimization

2019 ◽  
Vol 2019 ◽  
pp. 1-17
Author(s):  
Peng Xu ◽  
Xiaoming Wu ◽  
Man Guo ◽  
Shuai Wang ◽  
Qingya Li ◽  
...  
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Ideal Point ◽  
Dynamic Environment ◽  
Decision Space ◽  
Center Point ◽  
Multiobjective Optimization Problems ◽  
Objective Space ◽  
Adaptive Diversity ◽  
Dynamic Multiobjective Optimization

There are many issues to consider when integrating 5G networks and the Internet of things to build a future smart city, such as how to schedule resources and how to reduce costs. This has a lot to do with dynamic multiobjective optimization. In order to deal with this kind of problem, it is necessary to design a good processing strategy. Evolutionary algorithm can handle this problem well. The prediction in the dynamic environment has been the very challenging work. In the previous literature, the location and distribution of PF or PS are mostly predicted by the center point. The center point generally refers to the center point of the population in the decision space. However, the center point of the decision space cannot meet the needs of various problems. In fact, there are many points with special meanings in objective space, such as ideal point and CTI. In this paper, a hybrid prediction strategy carried through from both decision space and objective space (DOPS) is proposed to handle all kinds of optimization problems. The prediction in decision space is based on the center point. And the prediction in objective space is based on CTI. In addition, for handling the problems with periodic changes, a kind of memory method is added. Finally, to compensate for the inaccuracy of the prediction in particularly complex problems, a self-adaptive diversity maintenance method is adopted. The proposed strategy was compared with other four state-of-the-art strategies on 13 classic dynamic multiobjective optimization problems (DMOPs). The experimental results show that DOPS is effective in dynamic multiobjective optimization.

Modified Sub-Population Based Heat Transfer Search Algorithm for Structural Optimization

2017 ◽  
Vol 8 (3) ◽  
pp. 1-23 ◽  
Author(s):  
Ghanshyam Tejani ◽  
Vimal Savsani ◽  
Vivek Patel
Keyword(s):  
Heat Transfer ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Optimal Solution ◽  
Population Based ◽  
Truss Optimization ◽  
Feasible Region ◽  
Frequency Constraints ◽  
Local Optimal Solution ◽  
Better Than

In this study, a modified heat transfer search (MHTS) algorithm is proposed by incorporating sub-population based simultaneous heat transfer modes viz. conduction, convection, and radiation in the basic HTS algorithm. However, the basic HTS algorithm considers only one of the modes of heat transfer for each generation. The multiple natural frequency constraints in truss optimization problems can improve the dynamic behavior of the structure and prevent undesirable vibrations. However, shape and size variables subjected to frequency constraints are difficult to handle due to the complexity of its feasible region, which is non-linear, non-convex, implicit, and often converging to the local optimal solution. The viability and effectiveness of the HTS and MHTS algorithms are investigated by six standard trusses problems. The solutions illustrate that the MHTS algorithm performs better than the HTS algorithm.

A Mixed-Integer Memetic Algorithm Applied to Batch Process Optimization

2013 ◽  
Vol 300-301 ◽  
pp. 645-648 ◽  
Author(s):  
Yung Chien Lin
Keyword(s):  
Memetic Algorithm ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Search Space ◽  
Memetic Algorithms ◽  
Batch Processes ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Genetic Operators ◽  
Search Methods

Evolutionary algorithms (EAs) are population-based global search methods. Memetic Algorithms (MAs) are hybrid EAs that combine genetic operators with local search methods. With global exploration and local exploitation in search space, MAs are capable of obtaining more high-quality solutions. On the other hand, mixed-integer hybrid differential evolution (MIHDE), as an EA-based search algorithm, has been successfully applied to many mixed-integer optimization problems. In this paper, a mixed-integer memetic algorithm based on MIHDE is developed for solving mixed-integer constrained optimization problems. The proposed algorithm is implemented and applied to the optimal design of batch processes. Experimental results show that the proposed algorithm can find a better optimal solution compared with some other search algorithms.

The Use of Factorial Design to Improve a Harmony Search Algorithm to Synthesize Heat-Integrated Distillation Sequences

2018 ◽  
Vol 777 ◽  
pp. 218-225
Author(s):  
Somboon Sukpancharoen ◽  
Thongchai Rohitatisha Srinophakun ◽  
Jongjit Hirunlabh ◽  
Nopporn Rattanachoung
Keyword(s):  
Objective Function ◽  
Factorial Design ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Harmony Search ◽  
Harmony Search Algorithm ◽  
Mixed Integer ◽  
Integer Number ◽  
Flow Sheet ◽  
Stochastic Algorithm

Optimization problems often involve a large number of design variables, and the exact influence of each of these variables upon the objective function can become rather complex; there may exist local optima for the objective function, but for the typical heat-integrated distillation sequence, the matter of interest is solely the global optimum. Therefore, it is necessary to create a stochastic algorithm method which can synthesize distillation systems with multiple components. The encoding process employs and integer number series which allows the system flow sheet structure to be portrayed and then managed. Within this portrayal, the broad synthesis problem takes the form of an implicit MILP (mixed-integer linear programming) problem. This study considers the attributes of six well-known optimization algorithms: Harmony Search algorithm (HS), Artificial Bee Colony (ABC), Bat Algorithm (BA), Crow Search Optimization (CSO), Grew Wolf Optimization (GWO) and Monarch Butterfly Optimization (MBO). The optimal variables which influence the harmony search algorithm can be determined through full factorial design analysis. These variables can then be employed in the search to discover the optimal heat-integrated distillation sequence. The study investigates the attributes of the optimal configuration solution, in terms of harmony size (HS), required number of iterations, harmony memory considering the rate (HMCR), and pitch adjustment rate (PAR). The study then adopts the HS algorithm which is duly improved in order to address the problem. In comparison with alternative techniques, HS is more effective and more robust than other approaches.

A Smart Structural Algorithm (SSA) Based on Infeasible Region to Solve Mixed Integer Problems

2017 ◽  
Vol 8 (1) ◽  
pp. 24-44 ◽  
Author(s):  
Mohammad Hassan Salmani ◽  
Kourosh Eshghi
Keyword(s):  
Heuristic Algorithm ◽  
Feasible Solution ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Feasible Region ◽  
Limited Area ◽  
Objective Functions ◽  
Fields Of Study ◽  
Best Value

Optimization is an important fields of study in science where researchers seek to make the best and most practical decisions. Solving real optimization problems is an intractable issue which calls for generating an approximate using meta-heuristic algorithms. This study proposes a meta-heuristic algorithm which mainly searches the infeasible region. In this approach, the authors start from an infeasible solution, and while they try to get near to the feasible region, they ensure that the best value is kept for the objective function. The algorithm examines the space in such terms as Infeasibility and Objective Functions, Neighborhood Limited Area, Random Smart Points, and the calculation of new solutions. The algorithm can convert an infeasible solution to an appropriate corresponding feasible solution by applying a simple mathematical methodology. Finally, to test the efficiency of our algorithm, a sample random MIP problem and a hard benchmark TSP instance are solved and discussed in detail.

Mixed-integer nonlinear optimization

Acta Numerica ◽  
2013 ◽  
Vol 22 ◽  
pp. 1-131 ◽  
Author(s):  
Pietro Belotti ◽  
Christian Kirches ◽  
Sven Leyffer ◽  
Jeff Linderoth ◽  
James Luedtke ◽  
...  
Keyword(s):  
Branch And Bound ◽  
Convex Functions ◽  
State Of The Art ◽  
Small Sample ◽  
The State ◽  
Decision Problems ◽  
Mixed Integer ◽  
Feasible Region ◽  
Convex Relaxations ◽  
Single Tree

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

An Analysis of Fireworks Algorithm Solving Problems With Shifts in the Decision Space and Objective Space

2020 ◽  
pp. 277-311
Author(s):  
Shi Cheng ◽  
Junfeng Chen ◽  
Quande Qin ◽  
Yuhui Shi
Keyword(s):  
Problem Solving ◽  
Swarm Intelligence ◽  
Modular Arithmetic ◽  
Benchmark Problems ◽  
Search Range ◽  
Decision Space ◽  
Fireworks Algorithm ◽  
Objective Space ◽  
Single Objective

Fireworks algorithms for solving problems with the optima shifts in the decision space and/or objective space are analyzed. The standard benchmark problems have several weaknesses in the research of swarm intelligence algorithms for solving single-objective problems. The optimum shift in decision space and/or objective space will increase the difficulty of problem-solving. Modular arithmetic mapping is utilized in the original fireworks algorithm to handle solutions out of the search range. The solutions are implicitly guided to the center of search range for problems with symmetrical search range via this strategy. The optimization performance of the fireworks algorithm on shift functions may be affected by this strategy. Four kinds of mapping strategies are compared with different problems. The fireworks algorithms with mapping to the boundary or mapping to a limited stochastic region obtain good performance on problems with the optimum shift.

Multi-Objective Optimal Design of a PID Sliding Mode Controller With Three Different Reaching Laws

2019 ◽  
Author(s):  
Xiaotian Xu ◽  
Yousef Sardahi ◽  
Almuatazbellah Boker
Keyword(s):  
Optimal Design ◽  
Optimization Problems ◽  
Sliding Mode ◽  
Sliding Mode Controller ◽  
Reaching Law ◽  
Multi Objective ◽  
Conflicting Objectives ◽  
Objective Space ◽  
Pareto Fronts ◽  
Selection Of

Abstract Sliding mode controllers (SMCs) are well-known nonlinear control techniques. The design of a SMC involves the selection of a sliding mode surface and reaching law. The constant, exponential, and power rate reaching laws are the most widely used. Selecting a reaching law is often based on the desired reaching time; that is how fast the state trajectory approaches the switching manifold. However, the selection of a reaching law does not only affect the reaching time (tr) but also other design specifications such as the settling time (ts), overshoot (Mp), and tracking error (JIAE). Indeed, the design of a closed-loop system usually involves multiple and often conflicting objectives. Therefore, a multi-objective optimal design approach that takes into consideration all the design requirements should be adopted. Furthermore, a systematic study is needed to evaluate and compare the performance of a SMC controller under these reaching laws in multi-objective settings. To this end, the problems of designing a PID (Proportional-Integral-Derivative) sliding mode controller applied to linear and nonlinear dynamic systems using the three reaching laws are formulated as multi-objective optimization problems (MOPs). The objective space includes tr, Mp, ts, and JIAE and the parameter space consists of the design gains of the reaching laws and the sliding mode surface. The non-dominated sorting genetic algorithm (NSGA – II) is used to solve the optimization problem. The solution of the MOP is a Pareto front of optimal design points. Therefore, comparing three Pareto fronts is not a straightforward task. As a result, sections of the Pareto fronts that satisfy some legitimate constraints on the objective space are extracted. Then, a comparison among these sections is conducted graphically. The results show that the exponential rate reaching law outperforms the other two laws in most of the objectives under investigation.

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