scholarly journals Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

2016 ◽  
Vol 17 (1) ◽  
pp. 22-35 ◽  
Author(s):  
Dianzi Liu ◽  
Vassili Toropov
Keyword(s):  
Approximation Method ◽  
Optimization Problems ◽  
Continuous Optimization ◽  
Mixed Integer ◽  
Multipoint Approximation ◽  
Continuous Optimization Problems

Radius of Robust Feasibility for Mixed-Integer Problems

2021 ◽  
Author(s):  
Frauke Liers ◽  
Lars Schewe ◽  
Johannes Thürauf
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Continuous Optimization ◽  
Mixed Integer ◽  
Lp Relaxation ◽  
Maximal Size ◽  
Uncertainty Sets ◽  
Uncertainty Set ◽  
New Methodologies ◽  
Continuous Optimization Problems

For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal size of the uncertainty set such that robust feasibility of the MIP can be guaranteed. The approaches for the RRF in the literature are restricted to continuous optimization problems. We first analyze relations between the RRF of a MIP and its continuous linear (LP) relaxation. In particular, we derive conditions under which a MIP and its LP relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied to a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the RRF to MIPs and to include safe variables and constraints; that is, where uncertainties do not affect certain variables or constraints. In the extended setting, we again analyze relations between the RRF for a MIP and its LP relaxation. Afterward, we present methods for computing the RRF of LPs and of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF. Summary of Contribution: Robust optimization is an important field of operations research due to its capability of protecting optimization problems from data uncertainties that are usually defined via so-called uncertainty sets. Intensive research has been conducted in developing algorithmically tractable reformulations of the usually semi-infinite robust optimization problems. However, in applications it also important to construct appropriate uncertainty sets (i.e., prohibiting too conservative, intractable, or even infeasible robust optimization problems due to the choice of the uncertainty set). In doing so, it is useful to know the maximal “size” of a given uncertainty set such that a robust feasible solution still exists. In this paper, we study one notion of “size”: the radius of robust feasibility (RRF). We contribute on the theoretical side by generalizing the RRF to MIPs as well as to include “safe” variables and constraints (i.e., where uncertainties do not affect certain variables or constraints). This allows to apply the RRF to many applications since safe variables and constraints exist in most applications. We also provide first methods for computing the RRF of LPs as well as of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF.

scholarly journals Efficient hybrid algorithms to solve mixed discrete-continuous optimization problems

2018 ◽  
Vol 35 (2) ◽  
pp. 979-1002 ◽  
Author(s):  
Dianzi Liu ◽  
Chengyang Liu ◽  
Chuanwei Zhang ◽  
Chao Xu ◽  
Ziliang Du ◽  
...  
Keyword(s):  
Optimization Problems ◽  
Linear Optimization ◽  
Continuous Optimization ◽  
Trust Region ◽  
Adaptation Strategy ◽  
Hybrid Algorithms ◽  
Content Type ◽  
Multipoint Approximation ◽  
Discrete Values ◽  
Continuous Optimization Problems

PurposeIn real-world cases, it is common to encounter mixed discrete-continuous problems where some or all of the variables may take only discrete values. To solve these non-linear optimization problems, the use of finite element methods is very time-consuming. The purpose of this study is to investigate the efficiency of the proposed hybrid algorithms for the mixed discrete-continuous optimization and compare it with the performance of genetic algorithms (GAs).Design/methodology/approachIn this paper, the enhanced multipoint approximation method (MAM) is used to reduce the original nonlinear optimization problem to a sequence of approximations. Then, the sequential quadratic programing technique is applied to find the continuous solution. Following that, the implementation of discrete capability into the MAM is developed to solve the mixed discrete-continuous optimization problems.FindingsThe efficiency and rate of convergence of the developed hybrid algorithms outperforming GA are examined by six detailed case studies in the ten-bar planar truss problem, and the superiority of the Hooke–Jeeves assisted MAM algorithm over the other two hybrid algorithms and GAs is concluded.Originality/valueThe authors propose three efficient hybrid algorithms, the rounding-off, the coordinate search and the Hooke–Jeeves search-assisted MAMs, to solve nonlinear mixed discrete-continuous optimization problems. Implementations include the development of new procedures for sampling discrete points, the modification of the trust region adaptation strategy and strategies for solving mix optimization problems. To improve the efficiency and effectiveness of metamodel construction, regressorsfdefined in this paper can have the form in common with the empirical formulation of the problems in many engineering subjects.

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

1993 ◽  
Author(s):  
B. K. Kannan ◽  
Steven N. Kramer
Keyword(s):  
Lagrange Multiplier ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Optimization Problems ◽  
Mechanical Design ◽  
Continuous Optimization ◽  
Mixed Integer ◽  
Continuous Variables ◽  
Augmented Lagrange Multiplier

Abstract 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 & 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.

A Study of the Continuous Optimization Problem Using a Wood Robot Controlled by a Biologically Motivated System

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
Jong-Chen Chen
Keyword(s):  
Physical Environment ◽  
Intelligent System ◽  
Optimization Problems ◽  
Learning Algorithm ◽  
Continuous Optimization ◽  
Physical Structure ◽  
Experimental Results ◽  
Self Organized ◽  
Selection Operators ◽  
Continuous Optimization Problems

Continuous optimization plays an increasingly significant role in everyday decision-making situations. Our group had previously developed a multilevel system called the artificial neuromolecular system (ANM) that possessed structure richness allowing variation and/or selection operators to act on it in order to generate a broad range of dynamic behaviors. In this paper, we used the ANM system to control the motions of a wooden walking robot named Miky. The robot was used to investigate the ANM system's capability to deal with continuous optimization problems through self-organized learning. Evolutionary learning algorithm was used to train the system and generate appropriate control. The experimental results showed that Miky was capable of learning in a continued manner in a physical environment. A further experiment was conducted by making some changes to Miky's physical structure in order to observe the system's capability to deal with the change. Detailed analysis of the experimental results showed that Miky responded to the change by appropriately adjusting its leg movements in space and time. The results showed that the ANM system possessed continuous optimization capability in coping with the change. Our findings from the empirical experiments might provide us another dimension of information of how to design an intelligent system comparatively friendlier than the traditional systems in assisting humans to walk.

scholarly journals A Particle Swarm Based Algorithm for Functional Distributed Constraint Optimization Problems

2020 ◽  
Vol 34 (05) ◽  
pp. 7111-7118
Author(s):  
Moumita Choudhury ◽  
Saaduddin Mahmud ◽  
Md. Mosaddek Khan
Keyword(s):  
Optimization Problems ◽  
State Of The Art ◽  
Particle Swarm ◽  
Continuous Optimization ◽  
Continuous Variables ◽  
Constraint Optimization ◽  
Solution Quality ◽  
Distributed Constraint Optimization ◽  
Continuous Optimization Problems ◽  
Constraint Optimization Problems

Distributed Constraint Optimization Problems (DCOPs) are a widely studied constraint handling framework. The objective of a DCOP algorithm is to optimize a global objective function that can be described as the aggregation of several distributed constraint cost functions. In a DCOP, each of these functions is defined by a set of discrete variables. However, in many applications, such as target tracking or sleep scheduling in sensor networks, continuous valued variables are more suited than the discrete ones. Considering this, Functional DCOPs (F-DCOPs) have been proposed that can explicitly model a problem containing continuous variables. Nevertheless, state-of-the-art F-DCOPs approaches experience onerous memory or computation overhead. To address this issue, we propose a new F-DCOP algorithm, namely Particle Swarm based F-DCOP (PFD), which is inspired by a meta-heuristic, Particle Swarm Optimization (PSO). Although it has been successfully applied to many continuous optimization problems, the potential of PSO has not been utilized in F-DCOPs. To be exact, PFD devises a distributed method of solution construction while significantly reducing the computation and memory requirements. Moreover, we theoretically prove that PFD is an anytime algorithm. Finally, our empirical results indicate that PFD outperforms the state-of-the-art approaches in terms of solution quality and computation overhead.

Sign in / Sign up

Export Citation Format

Share Document