Decision Diagram Decomposition for Quadratically Constrained Binary Optimization

2020 ◽  
Author(s):  
David Bergman ◽  
Leonardo Lozano
Keyword(s):  
Benders Decomposition ◽  
Optimization Problems ◽  
State Of The Art ◽  
Cutting Plane Algorithm ◽  
Decision Diagrams ◽  
Quadratic Constraints ◽  
Binary Optimization ◽  
Limited Size ◽  
Quadratic Matrix ◽  
Quadratically Constrained

In recent years the use of decision diagrams within the context of discrete optimization has proliferated. This paper continues this expansion by proposing the use of decision diagrams for modeling and solving binary optimization problems with quadratic constraints. The model proposes the use of multiple decision diagrams to decompose a quadratic matrix so that each individual diagram has provably limited size. The decision diagrams are then linked through channeling constraints to ensure that the solution represented is consistent across the decision diagrams and that the original quadratic constraints are satisfied. The resulting family of decision diagrams are optimized over by a dedicated cutting-plane algorithm akin to Benders decomposition. The approach is general, in that commercial integer programming solvers can readily apply the technique. A thorough experimental evaluation on both benchmark and synthetic instances exhibits that the proposed decision diagram reformulation provides significant improvements over current methods for quadratic constraints in state-of-the-art solvers.

scholarly journals Automatic Design of Heuristic Algorithms for Binary Optimization Problems

2021 ◽  
Author(s):  
Marcelo de Souza
Keyword(s):  
Optimization Problem ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
State Of The Art ◽  
Algorithm Design ◽  
Automatic Design ◽  
Binary Optimization ◽  
Automatic Algorithm ◽  
Better Than ◽  
Design Techniques

In this work we present AutoBQP, a heuristic solver for binary optimization problems. It applies automatic algorithm design techniques to search for the best heuristics for a given optimization problem. Experiments show that the solver can find algorithms which perform better than or comparable to state-of-the-art methods, and can even find new best solutions for some instances of standard benchmark sets.

scholarly journals Investigation of Improved Cooperative Coevolution for Large-Scale Global Optimization Problems

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 146
Author(s):  
Aleksei Vakhnin ◽  
Evgenii Sopov
Keyword(s):  
Global Optimization ◽  
Evolutionary Algorithms ◽  
Numerical Experiments ◽  
Large Scale ◽  
Optimization Problems ◽  
State Of The Art ◽  
Fixed Number ◽  
High Dimensional ◽  
Cooperative Coevolution ◽  
Large Scale Problems

Modern real-valued optimization problems are complex and high-dimensional, and they are known as “large-scale global optimization (LSGO)” problems. Classic evolutionary algorithms (EAs) perform poorly on this class of problems because of the curse of dimensionality. Cooperative Coevolution (CC) is a high-performed framework for performing the decomposition of large-scale problems into smaller and easier subproblems by grouping objective variables. The efficiency of CC strongly depends on the size of groups and the grouping approach. In this study, an improved CC (iCC) approach for solving LSGO problems has been proposed and investigated. iCC changes the number of variables in subcomponents dynamically during the optimization process. The SHADE algorithm is used as a subcomponent optimizer. We have investigated the performance of iCC-SHADE and CC-SHADE on fifteen problems from the LSGO CEC’13 benchmark set provided by the IEEE Congress of Evolutionary Computation. The results of numerical experiments have shown that iCC-SHADE outperforms, on average, CC-SHADE with a fixed number of subcomponents. Also, we have compared iCC-SHADE with some state-of-the-art LSGO metaheuristics. The experimental results have shown that the proposed algorithm is competitive with other efficient metaheuristics.

A Novel Approach to Designing Surrogate-assisted Genetic Algorithms by Combining Efficient Learning of Walsh Coefficients and Dependencies

2021 ◽  
Vol 1 (2) ◽  
pp. 1-23
Author(s):  
Arkadiy Dushatskiy ◽  
Tanja Alderliesten ◽  
Peter A. N. Bosman
Keyword(s):  
Boolean Functions ◽  
Surrogate Model ◽  
Optimization Problems ◽  
State Of The Art ◽  
Fitness Function ◽  
Optimal Solution ◽  
Problem Structure ◽  
Novel Approach ◽  
Efficient Learning ◽  
Nk Landscapes

Surrogate-assisted evolutionary algorithms have the potential to be of high value for real-world optimization problems when fitness evaluations are expensive, limiting the number of evaluations that can be performed. In this article, we consider the domain of pseudo-Boolean functions in a black-box setting. Moreover, instead of using a surrogate model as an approximation of a fitness function, we propose to precisely learn the coefficients of the Walsh decomposition of a fitness function and use the Walsh decomposition as a surrogate. If the coefficients are learned correctly, then the Walsh decomposition values perfectly match with the fitness function, and, thus, the optimal solution to the problem can be found by optimizing the surrogate without any additional evaluations of the original fitness function. It is known that the Walsh coefficients can be efficiently learned for pseudo-Boolean functions with k -bounded epistasis and known problem structure. We propose to learn dependencies between variables first and, therefore, substantially reduce the number of Walsh coefficients to be calculated. After the accurate Walsh decomposition is obtained, the surrogate model is optimized using GOMEA, which is considered to be a state-of-the-art binary optimization algorithm. We compare the proposed approach with standard GOMEA and two other Walsh decomposition-based algorithms. The benchmark functions in the experiments are well-known trap functions, NK-landscapes, MaxCut, and MAX3SAT problems. The experimental results demonstrate that the proposed approach is scalable at the supposed complexity of O (ℓ log ℓ) function evaluations when the number of subfunctions is O (ℓ) and all subfunctions are k -bounded, outperforming all considered algorithms.

Nonlinear Optimal Design of Dynamic Mechanical Systems

1992 ◽  
Author(s):  
T. E. Potter ◽  
K. D. Willmert ◽  
M. Sathyamoorthy
Keyword(s):  
Objective Function ◽  
Optimization Problems ◽  
Iteration Process ◽  
Optimization Method ◽  
Linear Constraints ◽  
Sum Of Squares ◽  
Function Evaluation ◽  
Deformation Analysis ◽  
Nonlinear Constraints ◽  
Quadratic Constraints

Abstract Mechanism path generation problems which use link deformations to improve the design lead to optimization problems involving a nonlinear sum-of-squares objective function subjected to a set of linear and nonlinear constraints. Inclusion of the deformation analysis causes the objective function evaluation to be computationally expensive. An optimization method is presented which requires relatively few objective function evaluations. The algorithm, based on the Gauss method for unconstrained problems, is developed as an extension of the Gauss constrained technique for linear constraints and revises the Gauss nonlinearly constrained method for quadratic constraints. The derivation of the algorithm, using a Lagrange multiplier approach, is based on the Kuhn-Tucker conditions so that when the iteration process terminates, these conditions are automatically satisfied. Although the technique was developed for mechanism problems, it is applicable to any optimization problem having the form of a sum of squares objective function subjected to nonlinear constraints.

Solving Mono- and Multi-Objective Problems Using Hybrid Evolutionary Algorithms and Nelder-Mead Method

2021 ◽  
Vol 12 (4) ◽  
pp. 98-116
Author(s):  
Noureddine Boukhari ◽  
Fatima Debbat ◽  
Nicolas Monmarché ◽  
Mohamed Slimane
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Optimization Problems ◽  
State Of The Art ◽  
Hybrid Approach ◽  
Optimization Methods ◽  
Global Optimum ◽  
Stochastic Methods ◽  
Local Optima ◽  
Portfolio Optimization Problem

Evolution strategies (ES) are a family of strong stochastic methods for global optimization and have proved their capability in avoiding local optima more than other optimization methods. Many researchers have investigated different versions of the original evolution strategy with good results in a variety of optimization problems. However, the convergence rate of the algorithm to the global optimum stays asymptotic. In order to accelerate the convergence rate, a hybrid approach is proposed using the nonlinear simplex method (Nelder-Mead) and an adaptive scheme to control the local search application, and the authors demonstrate that such combination yields significantly better convergence. The new proposed method has been tested on 15 complex benchmark functions and applied to the bi-objective portfolio optimization problem and compared with other state-of-the-art techniques. Experimental results show that the performance is improved by this hybridization in terms of solution eminence and strong convergence.

scholarly journals Differential Evolution: A Survey and Analysis

2018 ◽  
Vol 8 (10) ◽  
pp. 1945 ◽  
Author(s):  
Tarik Eltaeib ◽  
Ausif Mahmood
Keyword(s):  
Global Optimization ◽  
Differential Evolution ◽  
Optimization Problems ◽  
State Of The Art ◽  
Population Based ◽  
Vital Role ◽  
Fine Tuning ◽  
Control Parameters ◽  
Hybrid Techniques ◽  
Global Optimizer

Differential evolution (DE) has been extensively used in optimization studies since its development in 1995 because of its reputation as an effective global optimizer. DE is a population-based metaheuristic technique that develops numerical vectors to solve optimization problems. DE strategies have a significant impact on DE performance and play a vital role in achieving stochastic global optimization. However, DE is highly dependent on the control parameters involved. In practice, the fine-tuning of these parameters is not always easy. Here, we discuss the improvements and developments that have been made to DE algorithms. In particular, we present a state-of-the-art survey of the literature on DE and its recent advances, such as the development of adaptive, self-adaptive and hybrid techniques.

scholarly journals Multi-Agent Systems Applications in Energy Optimization Problems: A State-of-the-Art Review

Energies ◽  
2018 ◽  
Vol 11 (8) ◽  
pp. 1928 ◽  
Author(s):  
Alfonso González-Briones ◽  
Fernando De La Prieta ◽  
Mohd Mohamad ◽  
Sigeru Omatu ◽  
Juan Corchado
Keyword(s):  
Optimization Problems ◽  
State Of The Art ◽  
Smart Cities ◽  
Energy Optimization ◽  
Sensor Data ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Research Perspectives ◽  
Wide Range ◽  
Multi Agent

This article reviews the state-of-the-art developments in Multi-Agent Systems (MASs) and their application to energy optimization problems. This methodology and related tools have contributed to changes in various paradigms used in energy optimization. Behavior and interactions between agents are key elements that must be understood in order to model energy optimization solutions that are robust, scalable and context-aware. The concept of MAS is introduced in this paper and it is compared with traditional approaches in the development of energy optimization solutions. The different types of agent-based architectures are described, the role played by the environment is analysed and we look at how MAS recognizes the characteristics of the environment to adapt to it. Moreover, it is discussed how MAS can be used as tools that simulate the results of different actions aimed at reducing energy consumption. Then, we look at MAS as a tool that makes it easy to model and simulate certain behaviors. This modeling and simulation is easily extrapolated to the energy field, and can even evolve further within this field by using the Internet of Things (IoT) paradigm. Therefore, we can argue that MAS is a widespread approach in the field of energy optimization and that it is commonly used due to its capacity for the communication, coordination, cooperation of agents and the robustness that this methodology gives in assigning different tasks to agents. Finally, this article considers how MASs can be used for various purposes, from capturing sensor data to decision-making. We propose some research perspectives on the development of electrical optimization solutions through their development using MASs. In conclusion, we argue that researchers in the field of energy optimization should use multi-agent systems at those junctures where it is necessary to model energy efficiency solutions that involve a wide range of factors, as well as context independence that they can achieve through the addition of new agents or agent organizations, enabling the development of energy-efficient solutions for smart cities and intelligent buildings.

scholarly journals Solving Delete Free Planning with Relaxed Decision Diagram Based Heuristics

2020 ◽  
Vol 67 ◽  
pp. 607-651
Author(s):  
Margarita Paz Castro ◽  
Chiara Piacentini ◽  
Andre Augusto Cire ◽  
J. Christopher Beck
Keyword(s):  
Linear Programming ◽  
Empirical Analysis ◽  
State Of The Art ◽  
The State ◽  
Decision Diagrams ◽  
Binary Decision ◽  
Admissible Heuristics ◽  
The Cost ◽  
Construction Algorithms ◽  
Diagram Representation

We investigate the use of relaxed decision diagrams (DDs) for computing admissible heuristics for the cost-optimal delete-free planning (DFP) problem. Our main contributions are the introduction of two novel DD encodings for a DFP task: a multivalued decision diagram that includes the sequencing aspect of the problem and a binary decision diagram representation of its sequential relaxation. We present construction algorithms for each DD that leverage these different perspectives of the DFP task and provide theoretical and empirical analyses of the associated heuristics. We further show that relaxed DDs can be used beyond heuristic computation to extract delete-free plans, find action landmarks, and identify redundant actions. Our empirical analysis shows that while DD-based heuristics trail the state of the art, even small relaxed DDs are competitive with the linear programming heuristic for the DFP task, thus, revealing novel ways of designing admissible heuristics.

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