scholarly journals A Comparative Study of Redundant Constraints Identification Methods in Linear Programming Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Paulraj S. ◽  
Sumathi P.
Keyword(s):  
Linear Programming ◽  
Large Scale ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Computational Effort ◽  
Linear Functions ◽  
Redundant Constraints ◽  
Computer Scientists ◽  
Constraints Solving ◽  
Linear Programming Problems

The objective function and the constraints can be formulated as linear functions of independent variables in most of the real-world optimization problems. Linear Programming (LP) is the process of optimizing a linear function subject to a finite number of linear equality and inequality constraints. Solving linear programming problems efficiently has always been a fascinating pursuit for computer scientists and mathematicians. The computational complexity of any linear programming problem depends on the number of constraints and variables of the LP problem. Quite often large-scale LP problems may contain many constraints which are redundant or cause infeasibility on account of inefficient formulation or some errors in data input. The presence of redundant constraints does not alter the optimal solutions(s). Nevertheless, they may consume extra computational effort. Many researchers have proposed different approaches for identifying the redundant constraints in linear programming problems. This paper compares five of such methods and discusses the efficiency of each method by solving various size LP problems and netlib problems. The algorithms of each method are coded by using a computer programming language C. The computational results are presented and analyzed in this paper.

A New Dynamic Basis Algorithm for Solving Linear Programming Problems for Engineering Design

1988 ◽  
Author(s):  
Y. Wang ◽  
E. Sandgren
Keyword(s):  
Linear Programming ◽  
Simplex Method ◽  
Inequality Constraints ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Active Constraint ◽  
Design Variables ◽  
Artificial Variable ◽  
Linear Programming Problems ◽  
Revised Simplex Method

Abstract A new linear programming algorithm is proposed which has significant advantages compared to the traditional simplex method. The search direction generated which is always along a common edge of the active constraint set, is used to locate candidate constraints, and can be used to modify the current basis. The dimension of the basis begins at one and dynamically increases but remains less than or equal to the number of design variables. This is true regardless of the number of inequality constraints present including upper and lower bounds. The proposed method can operate equally well from a feasible or infeasible point. The pivot operation and artificial variable strategy of the simplex method are not used. Examples are presented and results are compared with a traditional revised simplex method.

scholarly journals Benders decomposition with adaptive oracles for large scale optimization

2020 ◽  
Author(s):  
Nicolò Mazzi ◽  
Andreas Grothey ◽  
Ken McKinnon ◽  
Nagisa Sugishita
Keyword(s):  
Large Scale ◽  
Benders Decomposition ◽  
Optimization Problems ◽  
Cutting Plane ◽  
Computational Effort ◽  
Planning Problem ◽  
Illustrative Case ◽  
Investment Planning ◽  
Cutting Plane Algorithms ◽  
Inexact Information

AbstractThis paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an $$\epsilon $$ ϵ -optimal solution: an illustrative case is 31.9 times faster for a $$1.00\%$$ 1.00 % convergence tolerance and 15.4 times faster for a $$0.01\%$$ 0.01 % tolerance.

scholarly journals Convergence Analysis on an Accelerated Proximal Point Algorithm for Linearly Constrained Optimization Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Sha Lu ◽  
Zengxin Wei
Keyword(s):  
Machine Learning ◽  
Linear Programming ◽  
Proximal Point Algorithm ◽  
Original Problem ◽  
Optimization Problems ◽  
Linear Constraints ◽  
Proximal Point ◽  
Constrained Optimization Problems ◽  
Linearly Constrained ◽  
Linear Programming Problems

Proximal point algorithm is a type of method widely used in solving optimization problems and some practical problems such as machine learning in recent years. In this paper, a framework of accelerated proximal point algorithm is presented for convex minimization with linear constraints. The algorithm can be seen as an extension to G u ¨ ler’s methods for unconstrained optimization and linear programming problems. We prove that the sequence generated by the algorithm converges to a KKT solution of the original problem under appropriate conditions with the convergence rate of O 1 / k 2 .

Solving Large-Scale Zero-One Linear Programming Problems

1983 ◽  
Vol 31 (5) ◽  
pp. 803-834 ◽  
Author(s):  
Harlan Crowder ◽  
Ellis L. Johnson ◽  
Manfred Padberg
Keyword(s):  
Linear Programming ◽  
Large Scale ◽  
Linear Programming Problems
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