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Questions and Answers
Consider the dual of the following linear programming problem:
Max z = cTx subject to Ax = b, x ≥ 0, Let x* be a candidate solution for the primal problem.
Verify that if y* satisfies the constraints Ax* = b and x* ≥ 0 then it is feasible for the dual problem.
Consider the dual of the following linear programming problem:
Max z = cTx subject to Ax = b, x ≥ 0, Let x* be a candidate solution for the primal problem.
Verify that if y* satisfies the constraints Ax* = b and x* ≥ 0 then it is feasible for the dual problem.
If Ax* = b and x* ≥ 0, then it indicates the candidate solution (x*) satisfies the primal problem's constraints. Since the dual problem's constraints are derived from the primal problem's constraints, it follows that y* satisfying Ax* = b and x* ≥ 0 would also satisfy the dual problem's constraints. Hence, y* is feasible for the dual problem.
Consider the dual of the following linear programming problem:
Max z = cTx subject to Ax = b, x ≥ 0, Let x* be a candidate solution for the primal problem.
If x is not feasible, what condition must be violated in order for x to be dual infeasible?
Consider the dual of the following linear programming problem:
Max z = cTx subject to Ax = b, x ≥ 0, Let x* be a candidate solution for the primal problem.
If x is not feasible, what condition must be violated in order for x to be dual infeasible?
If x is not feasible, it means that it violates at least one of the constraints in the primal problem. The violation of these constraints directly leads to the violation of the dual problem's constraints, making the dual problem infeasible. The specific condition that would be violated, when x is not feasible, depends on the type of constraint and the specific violation.
Consider the dual of the following linear programming problem:
Min w=by subject to A y ≥ c, y ≥ 0, Let y* be a candidate solution for the dual problem.
Verify that if y* satisfies the constraints A y* ≥ c and y* ≥ 0 then it is feasible for the dual problem.
Consider the dual of the following linear programming problem:
Min w=by subject to A y ≥ c, y ≥ 0, Let y* be a candidate solution for the dual problem.
Verify that if y* satisfies the constraints A y* ≥ c and y* ≥ 0 then it is feasible for the dual problem.
If y* satisfies A y* ≥ c and y* ≥ 0, it indicates that the candidate solution (y*) satisfies the dual problem's constraints. Consequently, it is feasible for the dual problem.
Consider the dual of the following linear programming problem:
Min w=by subject to A y ≥ c, y ≥ 0, Let y* be a candidate solution for the dual problem.
If y* is not feasible, what condition must be violated in order for y* to be dual infeasible?
Consider the dual of the following linear programming problem:
Min w=by subject to A y ≥ c, y ≥ 0, Let y* be a candidate solution for the dual problem.
If y* is not feasible, what condition must be violated in order for y* to be dual infeasible?
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Explain the difference between the Weak Duality Theorem and the Strong Duality Theorem. Provide an example to illustrate each concept.
Explain the difference between the Weak Duality Theorem and the Strong Duality Theorem. Provide an example to illustrate each concept.
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Study Notes
Optimization Methods of Operations Research - Assignment 1
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Problem 1: Consider the dual of a linear programming problem.
- Verify feasibility of a candidate solution (x*) for the primal problem in the dual (if y* satisfies Ax* = b and x* ≥ 0, then y* is feasible for the dual).
- If primal solution x* is infeasible, the conditions that must be violated for it to be dual infeasible must be identified.
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Problem 2: Consider the dual of a linear programming problem.
- Verify feasibility of a candidate solution (y*) for the dual problem. (If y* satisfies Ay ≥ c and y* ≥ 0, then y* is feasible for the dual).
- If dual solution y* is infeasible, the conditions that must be violated for it to be dual infeasible must be identified.
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Problem 3: Weak vs. Strong Duality Theorems:
- Explain the differences between the weak and strong duality theorems.
- Provide example illustrations for each.
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Problem 4: Prove Weak Duality Theorem.
- Provide primal and dual programs as examples (max cx subject to Ax < b, x ≥ 0; min by subject to A*y > c, y ≥ 0).
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Problem 5: Prove Strong Duality Theorem.
- Prove the theorem for a primal-dual pair, assuming the Slater's condition holds.
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Problem 6: Complementary Slackness in Optimality Verification:
- Discuss the role of complementary slackness in verifying optimality for primal and dual solutions.
- Determine whether given primal and dual solutions satisfy the complementary slackness condition. Example provided: Primal: x₁ = 2, x₂ = 1; Dual: y₁ = 1.5, y₂ = 0.
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Problem 7: Dual Feasibility and Primal Feasibility: explain the concept of dual feasibility and primal feasibility, and how they relate to the duality theorems.
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Problem 8: Solve the dual problem explicitly given the primal problem and solution (provided in assignment prompt as a specific example.)
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Problem 9: Sensitivity Analysis and Dual Problems: explain how sensitivity analysis in linear programming relates to the dual problem.
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Problem 10: Karush-Kuhn-Tucker (KKT) for Dual Variables: Use the KKT conditions to find the dual variables for a given optimization problem.
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Problem 11: Primal Problem to Solve: Solve the primal and dual problems for a given primal problem (includes the objective function, constraints, and variables)
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Problem 12: Solving the dual problem explicitly, and verifying primal/dual optimal equality using Complementary Slackness when both the primal and dual problems are infeasible, explaining the implications for objective function modifications and significance of slackness for dual variables. Also, discussing the impact of the Strong Duality Theorem in real-world applications like resource allocation or transportation.
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Problem 13: Lagrange Multipliers: Using Lagrange multipliers to find critical points and determine their nature (maximum or minimum) given a constrained optimization problem. This problem explicitly provides an objective function (f(x, y)) and a constraint (g(x, y) = 0).
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Problem 14: Saddle Points: Find critical points of a function and investigate whether they are saddle points by applying the second-order derivative test, and displaying the geometric interpretation of a saddle point at (0,0)
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Problem 15: Critical Points and Global Extrema, determining critical points, and determining whether they represent local maximums, minimums or saddle points, and identifying global extrema over a given region.
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Problem 16: Dual of a Problem and KKT conditions: find the dual of a problem, use KKT to find the optimal solutions for both primal and dual problems, and discuss a relationship between their optimal values.
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Problem 17: Piecewise Function and Convexity: Investigate convexity properties of a piecewise function with a boundary condition. Examine whether the joining point of the given piecewise function causes non-differentiability. Discuss the impact on the function's convexity when the joining point is changed.
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Problem 18: Linear Program in Standard Form: formulate the dual, prove that the optimal values of primal and dual problems are equal under the Strong Duality Theorem, apply complementary slackness conditions to find solutions.
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Description
This assignment delves into the duality concepts in linear programming with a focus on verifying feasibility of candidate solutions for both primal and dual problems. Additionally, it discusses the Weak and Strong Duality Theorems, highlighting their differences and providing illustrative examples. Students will also be tasked with proving the Weak Duality Theorem.
