4.4 Revised simplex method
3 min read•july 30, 2024
The is a game-changer for solving big linear programming problems. It's like upgrading from a bicycle to a sports car – faster, more efficient, and better at handling complex situations. This method cuts down on data storage and reduces errors, making it a go-to for tackling large-scale optimization challenges.
At its core, the revised simplex method uses smart matrix operations to find the best solution. It's all about maintaining the right information, updating it efficiently, and making calculated decisions at each step. This approach not only speeds up the process but also opens doors to advanced techniques for even better performance.
Revised Simplex Method Motivation
Computational Efficiency and Data Management
Top images from around the web for Computational Efficiency and Data Management
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
1 of 3
Top images from around the web for Computational Efficiency and Data Management
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
3.3c. Examples – Simplex Method | Finite Math View original
Is this image relevant?
1 of 3
- Revised simplex method improves computational efficiency for large-scale linear programming problems
- Reduces data storage requirements by maintaining only essential information ( and objective row)
- Significantly reduces round-off errors accumulating in standard simplex method
- Particularly beneficial for problems with many variables and constraints
- Allows efficient handling of common in large-scale optimization problems
- Facilitates implementation of advanced techniques for selecting entering variables
- Partial pricing
- Steepest-edge pricing
Advanced Update Procedures
- Provides framework for implementing sophisticated update procedures
- Enables more efficient matrix operations and updates throughout optimization process
Applying the Revised Simplex Method
Matrix Operations and Variable Selection
- Uses matrix operations to compute necessary information at each
- Maintains and updates basis inverse () throughout optimization process
- Calculates for non-basic variables using formula
- Selects entering variable based on most negative reduced cost (maximization) or most positive reduced cost (minimization)
- Computes pivot column () to determine leaving variable using minimum ratio test
Optimality and Solution Calculation
- Reaches optimality when all reduced costs are non-negative (maximization) or non-positive (minimization)
- Obtains final by computing and setting all non-basic variables to zero
- Iteratively improves solution until optimality criteria are met
Basis Matrix Updates and Operations
Efficient Update Methods
- Updates basis inverse () at each iteration using efficient update formulas
- Employs Product Form of the Inverse (PFI) method to update
- Multiplies with elementary matrices representing row operations
- Utilizes Bartels-Golub update as alternative method
- Maintains factorization of basis matrix
- Often leads to improved numerical stability
Matrix Operations and Numerical Stability
- Implements efficient methods for solving systems of linear equations
- Applies partial pricing or multiple pricing to reduce computational effort in selecting entering variables
- Employs sparse matrix storage and manipulation techniques for handling large-scale problems
- Considers numerical stability through techniques like LU factorization with partial
- Maintains accuracy in matrix operations
Revised vs Standard Simplex Methods
Computational Approach and Efficiency
- Revised simplex method reduces storage requirements compared to standard simplex method
- Especially beneficial for large-scale problems with many variables and constraints
- Computes only necessary information at each iteration, unlike standard method maintaining entire tableau
- Exhibits better numerical stability and accuracy, particularly for problems prone to round-off errors
- Requires more complex implementation, involving sophisticated programming techniques
Advanced Features and Problem Suitability
- Provides natural framework for implementing advanced pivot selection rules and pricing strategies
- More challenging to implement in standard simplex method
- Both methods follow fundamental principles of moving between basic feasible solutions
- Revised simplex method preferred for solving large-scale linear programming problems
- Standard simplex method sufficient for smaller problems or educational purposes (classroom demonstrations)
Key Terms to Review (24)
Allowable Increase/Decrease: Allowable increase and decrease refer to the maximum amounts by which the coefficients of the objective function or the right-hand side constants in a linear programming problem can change without affecting the current optimal solution. These values provide crucial insights into how sensitive a solution is to changes in input parameters, allowing for informed decision-making in resource allocation and operational planning.
Bartels-Golub Update: The Bartels-Golub update is a numerical technique used to modify a matrix efficiently, particularly in the context of solving linear programming problems. This update is especially useful in the revised simplex method, where it allows for the quick adjustment of the inverse of a basis matrix when the basis changes, enhancing computational efficiency. By leveraging this update, one can avoid recalculating the entire inverse from scratch, making the optimization process faster and more efficient.
Basic Feasible Solution: A basic feasible solution is a solution to a linear programming problem that satisfies all the constraints and has a number of basic variables equal to the number of dimensions in the solution space. It represents a corner point or vertex of the feasible region, which is essential for understanding optimal solutions in linear programming. These solutions play a critical role in methods like the simplex algorithm, where they are iteratively refined to find the optimal solution.
Basis inverse: A basis inverse refers to the matrix that represents the inverse of a basis in a linear programming problem. This concept is pivotal in optimization, especially within the simplex algorithm framework, where it facilitates the transition between different feasible solutions. By understanding basis inverses, one can efficiently navigate the solution space and make necessary updates to the tableau during iterations.
Bland's Rule: Bland's Rule is a strategy used in the simplex method for linear programming that helps prevent cycling, which is when the algorithm revisits the same basic feasible solution repeatedly without making progress. This rule ensures that when choosing the next entering or leaving variable, the lowest indexed variable is selected, thereby providing a systematic approach to navigate through potential solutions. By implementing this rule, the revised simplex method can more effectively deal with cases of degeneracy, ensuring that the algorithm terminates successfully even when there are multiple optimal solutions or ties.
Constraint: A constraint is a condition or limitation that must be satisfied in an optimization problem. Constraints help define the feasible region within which solutions can be found, impacting how optimization algorithms like the simplex method operate and how certain cases like degeneracy or infeasibility arise in practice. Understanding constraints is essential because they directly influence the outcome of an optimization model and shape its practical applications across various fields.
Convergence: Convergence refers to the process by which an iterative algorithm approaches a final solution or optimal point as the number of iterations increases. This concept is critical in optimization methods, as it indicates the reliability and efficiency of algorithms in finding solutions. Understanding convergence helps assess the behavior of algorithms over time, ensuring that they provide accurate results while minimizing computational efforts.
Dual Simplex Method: The dual simplex method is an optimization technique used to solve linear programming problems by iterating on the dual feasible region while maintaining primal feasibility. It is particularly useful for problems where the primal solution becomes infeasible due to changes in constraints but where dual feasibility can still be achieved, such as in sensitivity analysis and problems involving resource allocation. This method connects deeply with the simplex algorithm and revised simplex method, highlighting special cases like degeneracy, unboundedness, and infeasibility, and extends into quadratic programming and semidefinite programming.
Feasible Solution: A feasible solution is a set of decision variables that satisfies all the constraints of an optimization problem. This concept is critical as it defines the boundaries within which optimal solutions can be found. Feasible solutions can vary in quality, and while some may be optimal, others may simply meet the necessary conditions without achieving the best possible outcome.
Iteration: Iteration refers to the repetitive process of refining a solution or approach to reach a desired outcome, often seen in optimization techniques. It involves repeatedly applying a specific set of operations or algorithms to progressively move closer to an optimal solution. In optimization methods like the simplex algorithm and its revisions, each iteration serves to improve the current solution by adjusting variables based on calculated pivots, ultimately converging toward an optimal feasible region.
LU Decomposition: LU decomposition is a mathematical method used to factor a matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U). This technique is crucial for solving systems of linear equations, inverting matrices, and calculating determinants efficiently, especially when dealing with larger matrices or iterative methods like the revised simplex method.
Matrix Inversion: Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix. This concept is crucial in linear algebra as it allows for solving systems of linear equations and plays a significant role in optimization techniques such as the revised simplex method, where efficient computations of basic feasible solutions are necessary to navigate the solution space.
Maximum Ratio Test: The maximum ratio test is a decision-making criterion used in the revised simplex method to determine the entering variable during an optimization process. It evaluates the potential increase in the objective function by calculating the ratio of the coefficients of the objective function to the coefficients of the constraints for each non-basic variable. This method helps identify which variable will provide the most significant improvement in the solution when it enters the basis.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, representing either a maximization or minimization task. It is typically formulated as a function of decision variables, which are the unknowns that need to be determined in order to achieve the best outcome based on given constraints.
Optimal Solution: An optimal solution refers to the best possible outcome or result for a given optimization problem, maximizing or minimizing an objective function while satisfying all constraints. Finding this solution is central to various mathematical modeling techniques, as it determines the most efficient or effective way to achieve goals under specified conditions.
Pivoting: Pivoting is a fundamental operation in linear programming that involves selecting a non-basic variable to enter the basis and a basic variable to leave the basis. This process allows for the improvement of the objective function by transforming the tableau, ultimately leading towards an optimal solution. It is crucial in both the iteration process of optimization and in methods like the revised simplex method, where efficient calculations are necessary to navigate through feasible solutions.
Product Form of the Inverse (PFI): The product form of the inverse is a mathematical technique used to simplify the calculation of the inverse of a matrix, especially in the context of linear programming. This approach allows for efficient updates to the inverse matrix during the iterative process of solving optimization problems, particularly within the revised simplex method. It leverages the properties of matrix operations to avoid recalculating the entire inverse from scratch, thus enhancing computational efficiency.
Reduced Costs: Reduced costs refer to the amount by which the objective function coefficient of a variable needs to be improved before that variable can enter the basis in a linear programming problem. Essentially, it measures how much the cost associated with an additional unit of a non-basic variable would affect the overall objective function. Understanding reduced costs is vital in various optimization techniques and helps in analyzing the potential impact of changing decision variables.
Revised Simplex Method: The revised simplex method is an optimization algorithm used to solve linear programming problems more efficiently than the standard simplex method. It focuses on updating only the necessary parts of the tableau during each iteration, which saves computational resources and makes it suitable for large-scale problems. This method employs matrix operations to maintain the solution efficiently, leveraging the basic feasible solution without having to recompute everything from scratch.
Shadow Price: Shadow price is the implicit value of a resource or constraint in optimization problems, representing the amount by which the objective function would improve if the resource were increased by one unit. This concept helps understand the trade-offs in resource allocation and the sensitivity of optimal solutions to changes in constraints. It provides insights into how scarce resources can be better utilized within a linear programming framework.
Sherman-Morrison Formula: The Sherman-Morrison formula provides a way to efficiently update the inverse of a matrix when that matrix undergoes a rank-one update. Specifically, if you have an invertible matrix and a vector, the formula gives a straightforward method to find the new inverse after modifying the original matrix by adding an outer product of two vectors. This is particularly useful in optimization methods where matrix inversions are frequent, such as in the revised simplex method, allowing for quicker calculations.
Slack Variable: A slack variable is an additional variable added to a linear programming problem to convert an inequality constraint into an equality constraint. By doing this, it allows for a more straightforward application of optimization techniques, as equality constraints are often easier to handle. Slack variables represent the difference between the left-hand side and right-hand side of a constraint, showing how much 'slack' there is in the available resources compared to the required limits.
Sparse Matrices: Sparse matrices are matrices in which most of the elements are zero. These matrices are crucial in optimization problems because they allow for efficient storage and computational methods, especially in large-scale applications where memory and processing power are limited. The use of sparse matrices can significantly speed up calculations and reduce the complexity of algorithms, making them particularly relevant in methods like the revised simplex method.
Transpose Matrix: A transpose matrix is formed by flipping a given matrix over its diagonal, effectively switching its rows with columns. This operation is essential in various mathematical techniques and optimization methods, as it helps in transforming equations and facilitating calculations. The transpose of a matrix is denoted by adding a superscript 'T' to the original matrix, which simplifies the representation of linear transformations and systems of equations.