Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Optimization → Linear Programming ↓


Simplex Method


The simplex method is an algorithm used to solve linear programming problems, which involves finding the maximum or minimum value of a linear function subject to linear constraints. Linear programming is an important optimization tool in various fields such as economics, engineering, military, business, and science.

Introduction to linear programming

Linear programming involves a set of equations and inequalities that represent a mathematical model whose optimal solution is sought. For a linear programming problem, the objective is to optimize (maximize or minimize) a linear objective function subject to linear equality and/or inequality constraints. The general form of a linear programming problem is given as:

Maximum (or minimum): c 1 x 1 + c 2 x 2 + ... + c n x n

subject to:
a 11 x 1 + a 12 x 2 + ... + a 1n x n ≤ b 1
a 21 x 1 + a 22 x 2 + ... + a 2n x n ≤ b 2,
a m1 x 1 + a m2 x 2 + ... + a mn x n ≤ b m

x 1 , x 2 , ..., x n ≥ 0

What is the simplex method?

The simplex method is an iterative procedure that systematically examines the corner points (or vertices) of the feasible region to find the optimal value of the objective function. This procedure involves moving from one vertex to another, improving the value of the objective function at each step, until the best possible value is achieved.

Simplex method procedure

  1. Convert the problem to standard form.
  2. Identify a viable initial solution.
  3. Keep working to find a better solution until no further improvement is possible.
  4. Assess and interpret the solution.

Step 1: Converting to standard form

To apply the simplex method, the linear programming problem must be transformed into the standard form. This involves:

  • Ensuring that all constraints are in the form of linear equations.
  • Introducing slack variables to convert inequalities into equalities.
  • Ensuring that all variables are non-negative.

Consider the following example:

Maximum: 3x 1 + 5x 2

subject to:
x 1 + 2x 2 ≤ 8
3x 1 + 2x 2 ≤ 12

x 1 , x 2 ≥ 0

Introduction to Slack Variables:

x 1 + 2x 2 + s 1 = 8
3x 1 + 2x 2 + s 2 = 12

x 1 , x 2 , s 1 , s 2 ≥ 0

The slack variables s 1 and s 2 represent unused resources from the constraints.

Step 2: Identify the initial solution

By setting the decision variables (here, x 1 and x 2 ) equal to zero, while allowing the slack variables to be equal to the constraint values, an initial basic feasible solution is identified.

In our example, the initial solution is:

x 1 = 0, x 2 = 0, s 1 = 8, s 2 = 12

Step 3: Iterate to improve the solution

To perform the recursion, a table is created. The objective row contains the negative coefficients of the objective function. The recursion involves:

  • The objective is to select the entering variable having the most negative coefficient in the row.
  • Selecting dropping variables using the minimum proportion test.
  • Pivoting to update the solution.
Initial tableau:
| basis | x 1 | x 2 | s 1 | s 2 | right hand side |
| S 1 | 1 | 2 | 1 | 0 | 8 |
| S 2 | 3 | 2 | 0 | 1 | 12 |
| z | -3 | -5 | 0 | 0 | 0 |

Iteration 1:

The entered variable is x 2 (most negative coefficient in the z-row: -5 ).

For minimum ratio: RHS / Coefficient of x 2 = min(8/2, 12/2) = 4, hence s 1 is the dropping variable.

Pivot to replace s 1 with x 2.

Pivoted Tableau:
| basis | x 1 | x 2 | s 1 | s 2 | right hand side |
| x 2 | 0.5 | 1 | 0.5 | 0 | 4 |
| S2 | 2.0 | 0 | -1 | 1 | 4 |
| z | -0.5 | 0 | 2.5 | 0 | 20 |

Iteration 2:

The most negative coefficient in the new z-line is -0.5 (for x 1 ).

Using min ratio: RHS / Coefficient of x 1 = min(4/0.5, 4/2) = 2, so s 2 is the dropping variable.

Pivot to replace s 2 with x 1.

Pivoted Tableau:
| basis | x 1 | x 2 | s 1 | s 2 | right hand side |
| x 2 | 0.5 | 1 | 0.5 | 0 | 4 |
| x1 | 1 | 0 | -0.5 | 0.5 | 2 |
| Z | 0 | 0 | 3 | 0.5 | 22 |

The value of the objective function has improved to 22, and there are no negative coefficients anymore, so the current solution is optimal.

Visual example

The shaded region in the graph represents the feasible region where all the constraints overlap. The circle marks the optimal solution point where the maximum value of the objective function is achieved.

Step 4: Assess and interpret the solution

The simplex method provides the best possible values for the decision variables within the feasible region. From the final table, we can determine the optimal solution:

x 1 = 2, x 2 = 4

Objective function value: 22

This means that the maximum value of the objective function 3x 1 + 5x 2 is 22 when x 1 = 2 and x 2 = 4.

Properties of the simplex method

  • The simplex method moves along the edge of the feasible region, ensuring that the solution improves or remains the same at each step until the optimal solution is found.
  • It efficiently handles linear objective functions with constraints, making it suitable for large-scale linear programming problems.
  • Although the method usually starts from the ground up, some preprocessing is still required to convert any problem into a standard form.
  • The algorithm is finite; it either converges to the optimal solution or recognizes that no solution exists for the given constraints.

Benefits and limitations

The advantages of the simplex method include its ability to provide exact optimal solutions to linear programming problems and its applicability to a wide range of problems beyond only constraints of the form ≤. Nevertheless, this method has some limitations:

Benefit

  • Provides accurate solution.
  • Can handle complex problems efficiently.
  • It is widely understood and used, and there are many libraries and software to support it.

Boundaries

  • Can be computationally expensive on very large datasets.
  • It doesn't work directly with non-linear problems.
  • Cycling can be problematic (sometimes, but this can be addressed with anti-cycling strategies).

Conclusion

The simplex method remains a powerful tool for optimization, especially in the field of operations research. It provides a structured approach to finding optimal solutions for problems that require maximization or minimization with linear constraints. Understanding both its strengths and limitations allows practitioners to effectively apply this method to solve real-world problems with reliability and accuracy.


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