Optimization Methods of Operations Research - Assignment 1
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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?

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.

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?

<p>If y* is not feasible, it means that it does not satisfy at least one of the constraints in the dual problem. Specifically, either A y* &lt; c or y* &lt; 0. This would make y* not a viable solution for the dual problem.</p> Signup and view all the answers

Explain the difference between the Weak Duality Theorem and the Strong Duality Theorem. Provide an example to illustrate each concept.

<p>The Weak Duality Theorem states that the optimal value of the dual problem is always less than or equal to the optimal value of the primal problem. However, the Strong Duality Theorem states that if the primal problem has a feasible solution and the dual problem has a feasible solution, then their optimal values are equal. For example, consider the following primal problem:</p> <p>Maximize z = 2x1 + x2 subject to</p> <p>x1 + x2 ≤ 4,</p> <p>2x1 + x2 ≤ 6</p> <p>x1, x2 ≥ 0.</p> <p>The dual of this problem is:</p> <p>Minimize w = 4y1 + 6y2 subject to</p> <p>y1 + 2y2 ≥ 2</p> <p>y1 + y2 ≥ 1</p> <p>y1, y2 ≥ 0</p> <p>Solving these problems, we find that the optimal value of the primal problem is 8 and the optimal value of the dual problem is also 8. This illustrates the Strong Duality Theorem. However, if the primal problem had a feasible solution, but the dual problem had no feasible solutions, then the Weak Duality Theorem would still apply. However, the optimal value of the dual problem would not be equal to the optimal value of the primal problem.</p> Signup and view all the answers

Flashcards

Dual Problem in Linear Programming

The dual problem seeks to maximize the objective function subject to constraints derived from the primal problem. The dual variables, often denoted as 'y', represent the shadow prices of the constraints in the primal problem.

Primal Problem in Linear Programming

The primal problem seeks to minimize the objective function subject to constraints that define the feasible region. The variables in the primal problem, often denoted as 'x', represent the decision variables.

Weak Duality Theorem

Weak duality states that the optimal value of the dual objective function is always less than or equal to the optimal value of the primal objective function. This holds true even if either problem is infeasible or unbounded.

Strong Duality Theorem

Strong duality states that if both the primal and dual problems have feasible solutions, then their optimal values are equal. This implies that the optimal solution of one problem can be obtained from the optimal solution of the other.

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Complementary Slackness

Complementary slackness is a key concept in duality theory. It states that for optimal solutions of both the primal and dual problems, the product of a primal variable and its corresponding dual constraint slack is zero. This implies that either a primal variable is zero or the corresponding dual constraint is tight (i.e., equality holds).

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Primal Feasibility

A solution is considered primal feasible if it satisfies all the constraints of the primal problem. This means that the chosen values for the primal variables fall within the feasible region defined by the constraints.

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Dual Feasibility

A solution is considered dual feasible if it satisfies all the constraints of the dual problem. In other words, the chosen values for the dual variables meet the requirements set forth in the dual problem.

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Sensitivity Analysis and Duality

Sensitivity analysis in linear programming explores how the optimal solution changes in response to variations in the problem parameters, such as objective function coefficients or constraint constants. The dual variables play a crucial role in sensitivity analysis as they represent the shadow prices, indicating how the optimal objective function value changes with changes in the constraints.

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KKT Conditions and Dual Variables

The Karush-Kuhn-Tucker (KKT) conditions are a set of necessary conditions for optimality in constrained optimization problems. They combine the Lagrangian function with constraint qualifications to identify potential optimal solutions. Dual variables are employed within the KKT conditions to account for the impact of constraints.

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Convex Function

A function is convex if the line segment connecting any two points on the function's graph lies above the graph. This means that for any two points x and y within the domain, and for any value of λ between 0 and 1, the following inequality holds: f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y).

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Saddle Point

A saddle point is a critical point of a multivariable function where the function is neither a local maximum nor a local minimum. It represents a point where the function increases in one direction and decreases in another direction.

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Method of Lagrange Multipliers

The method of Lagrange multipliers is a technique used in constrained optimization to find critical points. It introduces a Lagrange multiplier, denoted as 'λ', for each constraint, and then solves a system of equations formed by setting the gradient of the Lagrangian function, which includes the objective function and the constraints, equal to zero.

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Second-Order Conditions

The second-order conditions for multivariable functions are used to determine whether a critical point is a local maximum, local minimum, or saddle point. These conditions rely on the Hessian matrix, which consists of the second-order partial derivatives of the function. By analyzing the eigenvalues of the Hessian matrix, we can classify the critical points.

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Study Notes

Optimization Methods of Operations Research - Assignment 1

  • 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.

  • 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.

  • Problem 3: Weak vs. Strong Duality Theorems:

    • Explain the differences between the weak and strong duality theorems.
    • Provide example illustrations for each.

  • 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).

  • Problem 5: Prove Strong Duality Theorem.

    • Prove the theorem for a primal-dual pair, assuming the Slater's condition holds.

  • 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.

  • Problem 7: Dual Feasibility and Primal Feasibility: explain the concept of dual feasibility and primal feasibility, and how they relate to the duality theorems.

  • Problem 8: Solve the dual problem explicitly given the primal problem and solution (provided in assignment prompt as a specific example.)

  • Problem 9: Sensitivity Analysis and Dual Problems: explain how sensitivity analysis in linear programming relates to the dual problem.

  • Problem 10: Karush-Kuhn-Tucker (KKT) for Dual Variables: Use the KKT conditions to find the dual variables for a given optimization problem.

  • Problem 11: Primal Problem to Solve: Solve the primal and dual problems for a given primal problem (includes the objective function, constraints, and variables)

  • 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.

  • 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).

  • 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)

  • 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.

  • 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.

  • 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.

  • 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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MT 601 Optimization Methods of Operations Research Assignment 1 PDF

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.

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