CVL609 Lecture 4 Linear Programming II Dr Arnold Yuan, PEng Department of Civil Engineering
Agenda Duality of LP and Duality Simplex Method General Concept of Post-Optimality Analysis Sensitivity/Scenario Analysis of LP Remaining Issues in Simplex Method Initialization Unbounded feasible region (no solution) Empty feasible region (no solution) Multiple solutions
Learning Objectives After this lecture, students are expected to be able to Express the duality form of a given LP Choose the proper post-optimality analyses Use the final simplex table, and duality simplex if necessary, to perform sensitivity and scenario analyses Reference: Hillier and Liebermann (2015) chapter 6 & 8.1
Pt1 – Duality of LP
Review Simplex Method Decision variables and slack variables Basic variables and nonbasic variables Geometry Algebra CP solution Solution of n simultaneous equations Feasible solution Nonnegativeness of slack variables Move from one CPF to an adjacent CPF Exchange a basic and non-basic pair Optimality test Optimality test by checking the sign of coefficients in non- basic variables Determine moving direction: Steepest direction Determine entering variable: The nonbasic variable with the greatest improve rate Determine step size: Avoid infeasibility Determine leaving variable: The basic variable with the minimum ratio to avoid negative value Standard form of LP model Augmented/Slack form of LP model
A little stretch… Simplex Method The simplex iteration is equivalent to identify the set of nonbasic variables. • Once the nonbasic variables are identified, the basic variables can be solved by Gaussian elimination. • The identification of nonbasic variables is the same as the identification of active constraints. Example : Wyndows-Glass problem.
A Very Important Slide!!! A Fundamental Insight of LP Given y* and S* , the remaining simplex tableau can be established. Z* = y * b (Value of objective function) y * A – c ≥ 0 (Optimality) S * b ≥ 0 (Feasibility)
Example (cont’d) A I b S*A S* S* b y* y*A- c y* b -c
Duality of LP Primal Problem Dual Problem
Duality of LP Wyndow-Glass Example
Primal-Dual Relationships (Primal Feasible) (Dual Feasible) (Dual Infeasible) (Primal Infeasible) Complementary solutions property: At each iteration, the simplex method simultaneously identifies a CPF solution for the primal problem and a complementary solution for the dual problem, where . If x is not optimal for the primal problem, then is not feasible for the dual problem.
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- Spring '18
- Optimization, CPF, Empty