Linear Programming Summary

Optimization · Axiom Academy

SUMMARY Linear Programming Summary Let's review the key concepts from Unit 2: Linear Programming. Linear Objective: Maximize or minimize a linear function of decision variables Linear Constraints: All constraints are expressed as linear equations or inequalities Continuous Variables: Decision variables can take any real value (subject to constraints) Feasible Region: The set of all points satisfying all constraints forms a convex polygon Graphical Method: Plot constraints and identify corner points for 2D problems Simplex Method: Iterative algorithm that moves from corner to corner, improving the objective Any Dimension: Simplex works for problems with any number of variables and constraints Optimal at Corners: The optimal solution always occurs at a vertex of the feasible region Corner Point Theorem: If an optimal solution exists, it occurs at a corner point of the feasible region Duality: Every LP has an associated dual problem with complementary structure Strong Duality: If the primal has an optimal solution, so does the dual, with equal objective values Complementary Slackness: Relates primal and dual optimal solutions Example Recap: Simplex Iteration Step 1 - Choose Entering Variable: Select the variable with the most positive coefficient in the objective function (most negative reduced cost) Step 2 - Ratio Test: Compute ratios of right-hand side to entering variable coefficients for each constraint; choose the smallest non-negative ratio

This is the written version of the interactive lesson above. See the full Optimization course.