What is complementary slackness in linear programming

General
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what is complementary slackness in linear programming

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What is complementary slackness in linear programming?

Answer:

Complementary slackness is a critical concept in linear programming and optimization theory, particularly in the context of duality. It provides necessary and sufficient conditions for optimality in linear programming problems. Understanding complementary slackness helps in determining whether a given pair of primal and dual solutions is optimal.

Introduction to Linear Programming

In linear programming, we aim to optimize (either maximize or minimize) a linear objective function subject to a set of linear inequalities or equations, known as constraints. Each linear programming problem can be expressed in both a primal form and a dual form. The primal problem involves the original objective and constraints, while the dual problem is a derived problem with its own objective and constraints. The solutions to these two problems are related by duality theory.

Primal and Dual Problems

Consider a standard linear programming primal problem in the form of optimization of a function:

Maximize: (c^T x)
Subject to: (Ax \leq b)
(x \geq 0)

Where:

  • (c) is the coefficient vector for the objective function.
  • (x) is the vector of decision variables.
  • (A) is the matrix of coefficients for the constraints.
  • (b) is the vector of resource limits.

The dual problem would then be:

Minimize: (b^T y)
Subject to: (A^T y \geq c)
(y \geq 0)

Where:

  • (y) is the vector of dual variables or Lagrange multipliers.

Complementary Slackness Conditions

Complementary slackness provides a link between the primal and dual problems, outlining conditions that must hold for an optimal solution. The complementary slackness theorem states:

For a given primal solution (x^) and dual solution (y^), both must satisfy:

  1. ( (b - Ax^)^T y^ = 0 )
  2. ( (c - A^T y^)^T x^ = 0 )

Explanation:

  1. First Condition:
    This condition states that for each constraint of the primal problem, either the constraint is tight (i.e., (Ax = b)) or the corresponding dual variable (y_j) is zero. If a primal constraint has slack (i.e., is not tight), the dual variable associated with that constraint should not contribute to the objective in the dual problem.

  2. Second Condition:
    Similarly, for each dual constraint, either the constraint is tight (equality holds) or the corresponding primal variable (x_i) is zero. If a dual constraint is not active, it indicates that its corresponding primal variable should not contribute to the value of the primal objective.

Step 1: Present the Clues

Given the primal and dual problems, we have:

  • Primal variables: (x)
  • Dual variables: (y)
  • Primal constraints: (Ax \leq b)
  • Dual constraints: (A^T y \geq c)

Step 2: Deduction Process

By applying the complementary slackness conditions:

  • Identify which primal variables (x_i) are zero and check the corresponding dual constraints (A^T y \geq c) to ensure they are not tight.
  • For non-zero primal variables, verify the corresponding dual constraints are active.
  • Identify which dual variables (y_j) are zero and verify the corresponding primal constraints (Ax \leq b) are not tight.
  • For non-zero dual variables, verify that the corresponding primal constraints are active, meaning strict inequality does not hold.

Step 3: Finalize the Solution

Complementary slackness, when accompanied by feasible primal and dual solutions, provides strong evidence for optimality. It reduces the solution space by inferring the activity of constraints: if a constraint is active, it must contribute to the ultimate value, and if not, its contribution should be nullified through zero-valued variables.

Final Answer:

Complementary slackness is a set of conditions relating the primal and dual solutions in linear programming that assists in identifying optimal solutions. It states that for each pair of primal and dual solutions, either the slack in a primal constraint or the corresponding dual variable must be zero, and vice versa. This effectively assists in verifying the optimality of solutions within the theory of linear programming duality, functioning as an essential part of the duality theorem framework and providing practical utility in real-world optimization problems.