Improving the Optimality Verification and the Parallel Processing of the General Knapsack Linear Integer Problem

2021 ◽  
pp. 37-52
Author(s):  
Elias Munapo
Keyword(s):  
Parallel Processing ◽  
Objective Function ◽  
Branch And Bound ◽  
Equality Constraint ◽  
Search Region ◽  
New Approach ◽  
Integer Problem ◽  
Verification Process

The chapter presents a new approach to improve the verification process of optimality for the general knapsack linear integer problem. The general knapsack linear integer problem is very difficult to solve. A solution for the general knapsack linear integer problem can be accurately estimated, but it can still be very difficult to verify optimality using the brach and bound related methods. In this chapter, a new objective function is generated that is also used as a more binding equality constraint. This generated equality constraint can be shown to significantly reduce the search region for the branch and bound-related algorithms. The verification process for optimality proposed in this chapter is easier than most of the available branch and bound-related approaches. In addition, the proposed approach is massively parallelizable allowing the use of the much needed independent parallel processing.

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

2022 ◽  
Author(s):  
Vahid Mahmoodian ◽  
Iman Dayarian ◽  
Payman Ghasemi Saghand ◽  
Yu Zhang ◽  
Hadi Charkhgard
Keyword(s):  
Objective Function ◽  
Branch And Bound ◽  
Capacity Allocation ◽  
Branch And Cut ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Bargaining Solution ◽  
Reliability Optimization ◽  
Criterion Space ◽  
Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

Image Enhancement with Improved Global and Local Visual Effects

2021 ◽  
Vol 01 ◽  
Author(s):  
Muhammad Adeel ◽  
Yinglei Song
Keyword(s):  
Objective Function ◽  
Image Enhancement ◽  
Optimization Technique ◽  
Optimization Method ◽  
Original Image ◽  
Visual Effects ◽  
Swarm Optimization ◽  
New Approach ◽  
Visual Contrast ◽  
Global And Local

Background: In many applications of image processing, the enhancement of images is often a step necessary for their preprocessing. In general, for an enhanced image, the visual contrast as a whole and its refined local details are both crucial for achieving accurate results for subsequent classification or analysis. Objective: This paper proposes a new approach for image enhancement such that the global and local visual effects of an enhanced image can both be significantly improved. Methods: The approach utilizes the normalized incomplete Beta transform to map pixel intensities from an original image to its enhanced one. An objective function that consists of two parts is optimized to determine the parameters in the transform. One part of the objective function reflects the global visual effects in the enhanced image and the other one evaluates the enhanced visual effects on the most important local details in the original image. The optimization of the objective function is performed with an optimization technique based on the particle swarm optimization method. Results: Experimental results show that the approach is suitable for the automatic enhancement of images. Conclusion: The proposed approach can significantly improve both the global and visual contrasts of the image.

Solving Solid Transportation Problems With Multi-Choice Cost and Stochastic Supply and Demand

2018 ◽  
pp. 137-170 ◽  
Author(s):  
Sankar Kumar Roy ◽  
Deshabrata Roy Mahapatra
Keyword(s):  
Stochastic Programming ◽  
Objective Function ◽  
Transportation Problem ◽  
Supply And Demand ◽  
Optimal Solution ◽  
Programming Approach ◽  
Solid Transportation Problem ◽  
New Approach ◽  
Non Linear Programming ◽  
The Cost

In this chapter, the authors propose a new approach to analyze the Solid Transportation Problem (STP). This new approach considers the multi-choice programming into the cost coefficients of objective function and stochastic programming, which is incorporated in three constraints, namely sources, destinations, and capacities constraints, followed by Cauchy's distribution for solid transportation problem. The multi-choice programming and stochastic programming are combined into a solid transportation problem, and this new problem is called Multi-Choice Stochastic Solid Transportation Problem (MCSSTP). The solution concepts behind the MCSSTP are based on a new transformation technique that will select an appropriate choice from a set of multi-choice, which optimize the objective function. The stochastic constraints of STP converts into deterministic constraints by stochastic programming approach. Finally, the authors construct a non-linear programming problem for MCSSTP, and by solving it, they derive an optimal solution of the specified problem. A realistic example on STP is considered to illustrate the methodology.

Leader–follower-based dynamic trajectory planning for multirobot formation

Robotica ◽  
2013 ◽  
Vol 31 (8) ◽  
pp. 1351-1359 ◽  
Author(s):  
Shuang Liu ◽  
Dong Sun
Keyword(s):  
Analytical Solution ◽  
Objective Function ◽  
Collision Avoidance ◽  
Trajectory Planning ◽  
Trajectory Optimization ◽  
Optimal Trajectory ◽  
Dynamic Environment ◽  
Current Paper ◽  
New Approach ◽  
Dynamic Trajectory

SUMMARYThe present paper presents a new approach to a leader–follower-based dynamic trajectory planning for multirobot formation. A near-optimal trajectory is generated for each robot in a decentralized manner. The main contributions of the current paper are the proposal of a new objective function that considers both collision avoidance and formation requirement for the trajectory generation, and an analytical solution of trajectory parameters in the trajectory optimization. Simulations and experiments on multirobots are performed to demonstrate the effectiveness of the proposed approach to the multirobot formation in a dynamic environment.

New Approach for the Continuing Dynamic Programming

2012 ◽  
Vol 459 ◽  
pp. 575-578
Author(s):  
Peng Zhang ◽  
Xiang Huan Meng
Keyword(s):  
Dynamic Programming ◽  
Objective Function ◽  
Iteration Method ◽  
Programming Model ◽  
State Equations ◽  
New Approach ◽  
Optimal Value ◽  
Convergent Method ◽  
Optimal Values ◽  
Dynamic Programming Model

The paper proposes the discrete approximate iteration method to solve single-dimensional continuing dynamic programming model. The paper also presents a comparison of the discrete approximate iteration method and bi- convergent method to solve multi-dimensional continuing dynamic programming model. The algorithm is the following: Firstly, let state value of one of state equations be unknown and the others be known. Secondly, use discrete approximate iteration method to find the optimal value of the unknown state values, continue iterating until all state equations have found optimal values. If the objective function is convex, the algorithm is proved linear convergent. If the objective function is non-concave and non-convex, the algorithm is proved convergent.

Modeling and Simulation of Holonic Meta-Algorithm in Scheduling Process Using PERT Technique − Optimization in Job Shop Scheduling Problem

2015 ◽  
Vol 760 ◽  
pp. 199-204
Author(s):  
Mircea Gorgoi ◽  
Corneliu Neagu
Keyword(s):  
Objective Function ◽  
Job Shop ◽  
Job Shop Scheduling ◽  
Scheduling Problem ◽  
Job Shop Scheduling Problem ◽  
New Approach ◽  
Shop Scheduling ◽  
Technique Optimization ◽  
Optimum Solution

In generally scheduling can be viewed as optimization, bound by sequence and resource constrain and the minimization of the makespan is often used as the criterion. In this paper minimization of the makespan or complete time will be used such as an objective function and not the criterion of the decision. The new approach use heuristic elementary priority dispatch rules as the criterion of the decision. This research purpose a new methodology which use a specific elements of PERT techniques to find the optimum solution. New approach establish a solution's space where are find the all solution of the problem. Determination of the solution's space is realized by a meta-algorithm which take in account all the variant of the solutions of the process.

scholarly journals A New Approach to Find an Optimal Solution of a Fuzzy Linear Programming Problem by Fuzzy Dynamic Programming

2018 ◽  
Vol 7 (4.10) ◽  
pp. 360
Author(s):  
T. Nagalakshmi ◽  
G. Uthra
Keyword(s):  
Dynamic Programming ◽  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Fuzzy Linear Programming ◽  
New Approach ◽  
The Cost ◽  
Recursive Equations

This paper mainly focuses on a new approach to find an optimal solution of a fuzzy linear programming problem with the help of Fuzzy Dynamic Programming. Linear programming deals with the optimization of a function of variables called an objective function, subject to a set of linear inequalities called constraints. The objective function may be maximizing the profit or minimizing the cost or any other measure of effectiveness subject to constraints imposed by supply, demand, storage capacity, etc., Moreover, it is known that fuzziness prevails in all fields. Hence, a general linear programming problem with fuzzy parameters is considered where the variables are taken as Triangular Fuzzy Numbers. The solution is obtained by the method of FDP by framing fuzzy forward and fuzzy backward recursive equations. It is observed that the solutions obtained by both the equations are the same. This approach is illustrated with a numerical example. This feature of the proposed approach eliminates the imprecision and fuzziness in LPP models. The application of Fuzzy set theory in the field of dynamic Programming is called Fuzzy Dynamic Programming. 

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