5.3 Economic interpretation of duality

Duality theory in optimization connects resource allocation with economic value. Shadow prices, derived from dual variables, reveal how changes in constraints affect the objective function. This powerful tool helps decision-makers understand trade-offs and prioritize resources.

Primal and dual problems offer complementary perspectives on optimization. The relationship between their objective values, governed by duality theorems, enables efficient algorithms and provides insights into resource valuation and market equilibrium in various applications.

Dual Variables as Shadow Prices

Economic Interpretation of Dual Variables

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• Dual variables represent rate of change in objective function value with respect to changes in right-hand side constraints
• Shadow prices indicate marginal value of additional unit of resource
• Provide insights into sensitivity of optimal solution to changes in resource availability
• Correspond to optimal values of dual problem in linear programming
• Measure marginal contribution of each constraint to objective function

Applications of Dual Variables

• Reduced cost measures potential improvement in objective function by forcing non-basic variable into solution
• Determine which constraints are binding (active) at optimal solution
  • Non-zero dual variables indicate binding constraints
• Complementary slackness condition relates primal and dual variables
  • Either constraint is binding or associated dual variable is zero
• Used in sensitivity analysis to assess impact of resource changes (labor hours, raw materials)

Examples of Shadow Prices

• Manufacturing: $50 [shadow price](https://www.fiveableKeyTerm:Shadow_Price) for machine hours suggests increasing capacity by 1 hour increases profit by$50
• Agriculture: $100 shadow price for acre of land indicates additional acre increases revenue by$100
• Project management: 2-day shadow price for project deadline implies shortening deadline by 1 day increases cost by 2 units

Primal vs Dual Objective Values

Duality Theorems

• Weak Duality Theorem: Primal objective value ≤ Dual objective value for any feasible solutions
• Strong Duality Theorem: Primal and dual objective values equal at optimality
  • Requires certain conditions (Slater's condition for convex optimization)
• Duality gap: Difference between primal and dual objective values
  • Becomes zero at optimality for problems satisfying strong duality
• Complementary slackness conditions verify optimality of primal and dual solutions
  • Product of each primal variable and corresponding dual constraint slack equals zero

Applications in Optimization

• Relationship between primal and dual objectives enables development of efficient algorithms
  • Primal-dual method simultaneously solves both problems
• Integer programming uses integrality gap
  • Measures difference between optimal values of integer program and linear programming relaxation
• Economic interpretation highlights relationship between resource allocation (primal) and resource valuation (dual)

Examples of Primal-Dual Relationships

• Production planning: Primal maximizes profit, dual minimizes resource costs
• Network flow: Primal maximizes flow, dual minimizes cut capacity
• Portfolio optimization: Primal maximizes return, dual minimizes risk

Duality in Resource Allocation

Economic Framework

• Provides understanding of trade-offs between different resources and impact on overall objective
• Dual problem interprets as finding most efficient pricing scheme for resources
  • Maximizes total resource value while ensuring no activity is profitable
• Shadow prices indicate marginal value of resources
  • Help identify critical resources for system performance
• Opportunity cost closely related to dual variables
  • Represent cost of using resource in terms of foregone alternatives

Decision-Making Applications

• Sensitivity analysis using dual information evaluates impact of changes in resource availability or constraints
  • Assesses effects on optimal solution and overall system performance
• Economic interpretation of reduced costs determines which non-basic variables to consider for inclusion in optimal solution
  • Helps prioritize potential new products or activities
• Provides insights into market equilibrium
  • Primal problem represents production side
  • Dual problem represents consumption side of economy

Resource Allocation Examples

• Manufacturing: Dual variables show value of increasing machine capacity or labor hours
• Investment: Shadow prices indicate potential returns from allocating more capital to specific projects
• Supply chain: Duality analysis reveals most valuable transportation routes or warehouse locations