The Need To Visualize Sudoku

2021 ◽  
Author(s):  
Navin Ipe ◽  
Raghavendra V. Kulkarni
Keyword(s):  
Evolutionary Algorithms ◽  
Fitness Landscape ◽  
Optimal Solution ◽  
Brute Force ◽  
Problem Space ◽  
Evolutionary Methods ◽  
Hybrid Approaches ◽  
Mathematical Properties ◽  
Multidimensional Problems ◽  
Unique Method

<div>Sudoku puzzles are easily solved using backtracking algorithms. Yet, literature is scattered with various shy and often opaque attempts at using evolutionary algorithms and hybridizing them with known strategies of solving the puzzle. Evolutionary methods are serendipitous in nature, and this paper demonstrates the behaviour of such serendipity under constraints, using visuals that depict the sheer magnitude of the problem space and the nature in which intertwined constraints affect the scope for locating a solution, with the hope that it could inspire a new way of looking at the problem. We propose a method of visualizing the sudoku fitness landscape, the vastness and complexity of even partially brute forcing the puzzle, and a unique method of mutating puzzle states using circular swaps. These insights could potentially serve as a link to comprehend the problem space when designing solutions for vast, multidimensional problems. Additionally, finding the optimal solution for some puzzles was notably harder, compared to puzzles in the same category of given clues. A short investigation was conducted into this phenomenon, which revealed hints that compel us to propose that the direction of research that should be taken, is in discovering more about puzzle states and definitive mathematical properties of the puzzle, rather than merely designing brute-force, stochastic or hybrid approaches of finding solutions.</div>

scholarly journals The Need To Visualize Sudoku

2021 ◽  
Author(s):  
Navin Ipe ◽  
Raghavendra V. Kulkarni
Keyword(s):  
Evolutionary Algorithms ◽  
Fitness Landscape ◽  
Optimal Solution ◽  
Brute Force ◽  
Problem Space ◽  
Evolutionary Methods ◽  
Hybrid Approaches ◽  
Mathematical Properties ◽  
Multidimensional Problems ◽  
Unique Method

<div>Sudoku puzzles are easily solved using backtracking algorithms. Yet, literature is scattered with various shy and often opaque attempts at using evolutionary algorithms and hybridizing them with known strategies of solving the puzzle. Evolutionary methods are serendipitous in nature, and this paper demonstrates the behaviour of such serendipity under constraints, using visuals that depict the sheer magnitude of the problem space and the nature in which intertwined constraints affect the scope for locating a solution, with the hope that it could inspire a new way of looking at the problem. We propose a method of visualizing the sudoku fitness landscape, the vastness and complexity of even partially brute forcing the puzzle, and a unique method of mutating puzzle states using circular swaps. These insights could potentially serve as a link to comprehend the problem space when designing solutions for vast, multidimensional problems. Additionally, finding the optimal solution for some puzzles was notably harder, compared to puzzles in the same category of given clues. A short investigation was conducted into this phenomenon, which revealed hints that compel us to propose that the direction of research that should be taken, is in discovering more about puzzle states and definitive mathematical properties of the puzzle, rather than merely designing brute-force, stochastic or hybrid approaches of finding solutions.</div>

scholarly journals The Need to Visualize Sudoku

2021 ◽  
Author(s):  
Navin K Ipe
Keyword(s):  
Evolutionary Algorithms ◽  
Fitness Landscape ◽  
Optimal Solution ◽  
Brute Force ◽  
Problem Space ◽  
Evolutionary Methods ◽  
Hybrid Approaches ◽  
Mathematical Properties ◽  
Multidimensional Problems ◽  
Unique Method

Sudoku puzzles are easily solved using backtracking algorithms. Yet, literature is scattered with various shy and often opaque attempts at using evolutionary algorithms and hybridizing them with known strategies of solving the puzzle. Evolutionary methods are serendipitous in nature, and this paper demonstrates the behaviour of such serendipity under constraints, using visuals that depict the sheer magnitude of the problem space and the nature in which intertwined constraints affect the scope for locating a solution, with the hope that it could inspire a new way of looking at the problem. We propose a method of visualizing the sudoku fitness landscape, the vastness and complexity of even partially brute forcing the puzzle, and a unique method of mutating puzzle states using circular swaps. These insights could potentially serve as a link to comprehend the problem space when designing solutions for vast, multidimensional problems. Additionally, finding the optimal solution for some puzzles was notably harder, compared to puzzles in the same category of given clues. A short investigation was conducted into this phenomenon, which revealed hints that compel us to propose that the direction of research that should be taken, is in discovering more about puzzle states and definitive mathematical properties of the puzzle, rather than merely designing brute-force, stochastic or hybrid approaches of finding solutions.

Applied Process Simulation-Driven Oil and Gas Separation Plant Optimization using Surrogate Modeling and Evolutionary Algorithms

2019 ◽  
Author(s):  
Anders Andreasen
Keyword(s):  
Evolutionary Algorithms ◽  
Gas Separation ◽  
Oil And Gas ◽  
Optimal Solution ◽  
Computer Experiment ◽  
Latin Hypercube Sampling ◽  
Surrogate Models ◽  
Inlet Temperature ◽  
Separation Plant ◽  
Separation Stage

In this article the optimization of a realistic oil and gas separation plant has been studied. Two different fluids are investigated and compared in terms of the optimization potential. Using Design of Computer Experiment (DACE) via Latin Hypercube Sampling (LHS) and rigorous process simulations, surrogate models using Kriging have been established for selected model responses. The surrogate models are used in combination with a variety of different evolutionary algorithms for optimizing the operating profit, mainly by maximizing the recoverable oil production. A total of 10 variables representing pressure and temperature various key places in the separation plant are optimized to maximize the operational profit. The optimization is bounded in the variables and a constraint function is included to ensure that the optimal solution allows export of oil with an RVP < 12 psia. The main finding is that, while a high pressure is preferred in the first separation stage, apparently a single optimal setting for the pressure in downstream separators does not appear to exist. In the second stage separator apparently two different, yet equally optimal, settings are revealed. In the third and final separation stage a correlation between the separator pressure and the applied inlet temperature exists, where different combinations of pressure and temperature yields equally optimal results.<br>

A New Method for Modeling the Behavior of Finite Population Evolutionary Algorithms

2010 ◽  
Vol 18 (3) ◽  
pp. 451-489 ◽  
Author(s):  
Tatsuya Motoki
Keyword(s):  
Markov Chain ◽  
Evolutionary Algorithms ◽  
Finite Population ◽  
Fitness Landscape ◽  
Markov Chain Model ◽  
Search Space ◽  
Chain Model ◽  
Transition Matrices ◽  
Memory Space ◽  
Exact Model

As practitioners we are interested in the likelihood of the population containing a copy of the optimum. The dynamic systems approach, however, does not help us to calculate that quantity. Markov chain analysis can be used in principle to calculate the quantity. However, since the associated transition matrices are enormous even for modest problems, it follows that in practice these calculations are usually computationally infeasible. Therefore, some improvements on this situation are desirable. In this paper, we present a method for modeling the behavior of finite population evolutionary algorithms (EAs), and show that if the population size is greater than 1 and much less than the cardinality of the search space, the resulting exact model requires considerably less memory space for theoretically running the stochastic search process of the original EA than the Nix and Vose-style Markov chain model. We also present some approximate models that use still less memory space than the exact model. Furthermore, based on our models, we examine the selection pressure by fitness-proportionate selection, and observe that on average over all population trajectories, there is no such strong bias toward selecting the higher fitness individuals as the fitness landscape suggests.

scholarly journals Automation of the one-storey underground parking space organization

2021 ◽  
Vol 284 ◽  
pp. 06012
Author(s):  
Natalia Knyazeva ◽  
Anastasia Kolosova
Keyword(s):  
Evolutionary Algorithms ◽  
Urban Area ◽  
Optimal Solution ◽  
Parking Space ◽  
Space Planning ◽  
Parking Lots ◽  
Legal Documents ◽  
Space Organization ◽  
Big Cities ◽  
The One

With the growing car population in big cities, the problem of its keeping in conditions of a compact urban area has happened. The organisation of parking space in a different way has resolved this issue. Underground parking was in demand in many countries even in the XX century. By the way, they are becoming more and more popular now. The design of car parking is based on legal documents, which regulate the size of car parking seats and the width of the passage inside the garage. It is expedient to use evolutionary algorithms as one of the tools of algorithmic modelling for automation of design the car parking lots and for identifying the most effective and profitable way of the car parking space planning. So, the process of looking for the most optimal solution in underground car parking designing.

scholarly journals Multiple infill criterion-assisted hybrid evolutionary optimization for medium-dimensional computationally expensive problems

2021 ◽  
Author(s):  
Shufen Qin ◽  
Chan Li ◽  
Chaoli Sun ◽  
Guochen Zhang ◽  
Xiaobo Li
Keyword(s):  
Evolutionary Algorithms ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Global Search ◽  
Model Management ◽  
The Other ◽  
Local Searches ◽  
Fitness Value ◽  
Expensive Problems ◽  
Computationally Expensive

AbstractSurrogate-assisted evolutionary algorithms have been paid more and more attention to solve computationally expensive problems. However, model management still plays a significant importance in searching for the optimal solution. In this paper, a new method is proposed to measure the approximation uncertainty, in which the differences between the solution and its neighbour samples in the decision space, and the ruggedness of the objective space in its neighborhood are both considered. The proposed approximation uncertainty will be utilized in the surrogate-assisted global search to find a solution for exact objective evaluation to improve the exploration capability of the global search. On the other hand, the approximated fitness value is adopted as the infill criterion for the surrogate-assisted local search, which is utilized to improve the exploitation capability to find a solution close to the real optimal solution as much as possible. The surrogate-assisted global and local searches are conducted in sequence at each generation to balance the exploration and exploitation capabilities of the method. The performance of the proposed method is evaluated on seven benchmark problems with 10, 20, 30 and 50 dimensions, and one real-world application with 30 and 50 dimensions. The experimental results show that the proposed method is efficient for solving the low- and medium-dimensional expensive optimization problems by compared to the other six state-of-the-art surrogate-assisted evolutionary algorithms.

scholarly journals Performance of Resource Allocation in Device-to-Device Communication Systems Based on Evolutionally Optimization Algorithms

2018 ◽  
Author(s):  
Hsu-Tan Tan ◽  
Bor-An Chen ◽  
Yung-Fa Huang
Keyword(s):  
Particle Swarm Optimization ◽  
Evolutionary Algorithms ◽  
Communication Systems ◽  
Particle Swarm ◽  
Optimal Solution ◽  
System Capacity ◽  
Swarm Optimization ◽  
Device To Device ◽  
Simulation Results ◽  
Capacity Performance

In this study, the resource blocks (RB) are allocated to user equipment (UE) according to the evolutional algorithms for long term evolution (LTE) systems. Particle Swarm Optimization (PSO) algorithm is one of the evolutionary algorithms, based on the imitation of a flock of birds foraging behavior through learning and grouping the best experience. In previous work, the Simple Particle Swarm Optimization (SPSO) algorithm was proposed for RB allocation to enhance the throughput of Device-to-Device (D2D) communications and improve the system capacity performance. In simulation results, with less population size of M = 10, the SPSO can perform quickly convergence to sub-optimal solution in the 100th generation and obtained sub-optimum performance with more 2 UEs than the Rand method. Genetic algorithm (GA) is one of the evolutionary algorithms, based on Darwinian models of natural selection and evolution. Therefore, we further proposed a Refined PSO (RPSO) and a novel GA to enhance the throughput of UEs and to improve the system capacity performance. Simulation results show that the proposed GA with 100 populations, in 200 generations can converge to suboptimal solutions. Therefore, with comparing with the SPSO algorithm the proposed GA and RPSO can improve system capacity performance with 1.8 and 0.4 UEs, respectively.

Comparison of Multi-Objective Evolutionary Algorithms to Prioritize Regression Test Cases

2021 ◽  
Vol 9 (12) ◽  
pp. 1445-1454
Keyword(s):  
Fault Detection ◽  
Evolutionary Algorithms ◽  
Fitness Function ◽  
Optimal Solution ◽  
Test Suite ◽  
Test Case ◽  
Test Cases ◽  
Nsga Ii ◽  
Multi Objective ◽  
Regression Test

Regression testing is one of the most critical testing activities among software product verification activities. Nevertheless, resources and time constraints could inhibit the execution of a full regression test suite, hence leaving us in confusion on what test cases to run to preserve the high quality of software products. Different techniques can be applied to prioritize test cases in resource-constrained environments, such as manual selection, automated selection, or hybrid approaches. Different Multi-Objective Evolutionary Algorithms (MOEAs) have been used in this domain to find an optimal solution to minimize the cost of executing a regression test suite while obtaining maximum fault detection coverage as if the entire test suite was executed. MOEAs achieve this by selecting set of test cases and determining the order of their execution. In this paper, three Multi Objective Evolutionary Algorithms, namely, NSGA-II, IBEA and MoCell are used to solve test case prioritization problems using the fault detection rate and branch coverage of each test case. The paper intends to find out what’s the most effective algorithm to be used in test cases prioritization problems, and which algorithm is the most efficient one, and finally we examined if changing the fitness function would impose a change in results. Our experiment revealed that NSGA-II is the most effective and efficient MOEA; moreover, we found that changing the fitness function caused a significant reduction in evolution time, although it did not affect the coverage metric.

Evolutionary Algorithms for Global Decision Tree Induction

2018 ◽  
pp. 2132-2141 ◽  
Author(s):  
Marek Kretowski ◽  
Marcin Czajkowski
Keyword(s):  
Evolutionary Algorithms ◽  
Decision Tree ◽  
Knowledge Discovery ◽  
Decision Trees ◽  
Optimal Solution ◽  
Top Down ◽  
Local Optima ◽  
Optimal Decisions ◽  
Decision Tree Induction ◽  
Regression Functions

Decision trees represent one of the main predictive techniques in knowledge discovery. This chapter describes evolutionary induced trees, which are emerging alternatives to the greedy top-down solutions. Most typical tree-based system searches only for locally optimal decisions at each node and do not guarantee the optimal solution. Application of evolutionary algorithms to the problem of decision tree induction allows searching for the structure of the tree, tests in internal nodes and regression functions in the leaves (for model trees) at the same time. As a result, such globally induced decision tree is able to avoid local optima and usually leads to better prediction than the greedy counterparts.

scholarly journals Proximity measures based on KKT points for constrained multi-objective optimization

2020 ◽  
Author(s):  
Gabriele Eichfelder ◽  
Leo Warnow
Keyword(s):  
Evolutionary Algorithms ◽  
Optimal Solution ◽  
Necessary Optimality Conditions ◽  
Kkt Conditions ◽  
Test Problems ◽  
Numerical Verification ◽  
Termination Criteria ◽  
Continuity Properties ◽  
Optimal Solution Set ◽  
Proximity Measures

AbstractAn important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush–Kuhn–Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the violation of the KKT conditions. It can be computed easily and is continuous in every efficient solution. Hence, it can be used as an indicator for the proximity of a certain point to the set of efficient (Edgeworth-Pareto-minimal) solutions and is well suited for algorithmic use due to its continuity properties. This is especially useful within evolutionary algorithms for candidate selection and termination, which we also illustrate numerically for some test problems.

Sign in / Sign up

Export Citation Format

Share Document