duality and post optimal analysis

PDF	Chapter 4 Duality Suppose we wish to increase profit by expanding manufacturing capacities In such a situation it is useful to think of profit as a function of a vector ∆ ∈

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Lecture 10: Duality and Post-Optimal Analysis

Duality and Post-Optimal Analysis September 21 2009 Overview Dual of an LP Problem (4 1) Primal-Dual Relationship (4 2 1) Simplex Table Computations (4 2 2–4 2 4) GE 330/IE 310 1 LP Dual The original LP is referred to as primal The dual is another LP associated with the given LP It is defined systematically An LP and its dual are closely related:

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Play:block;margin-top:24px;margin-bottom:2px;\ class=\tit wwwifpillinoiseduLecture 11: Post-Optimal Analysis

•Using dual simplex method we can determine an optimal so-lution for the perturbed problem from the optimal table of the original problem •Using the dual problem/duality we can analyze additional changes •Introducing new operations (resources) LP Model: New constraints in the original (primal) problem •Introducing new products (activities)

Is there a finite optimal solution for both primal and dual problems?
If a finite feasible solution exists for both the primal and dual problems, then there exists a finite optimal solution for both problems where In other words, the maximum feasible value of the primal objective function equals the minimum feasible value of the dual objective function. Note then for any feasible solution,
Which theorem states One immediate implication of weak duality?
The following theorem states one immediate implication of weak duality. Theorem 4.2.2. (Certificate of Optimality) If x and y are feasible so-lutions of the primal and dual and cT x = bT y, then x and y must be optimal solutions to the primal and dual.
Does duality theory apply to linear programs?
Duality theory applies to general linear programs, that can involve greater-than, less-than, and equality constraints. However, to keep things simple, we will only study in this section linear programs that are in symmetric form. Such linear programs take the form: x ≥ 0. for some matrix A ∈
How do you find a dual objective?
Consider the following linear program (LP). Plot the feasible region of the primal and show that the primal objec-tive value goes to infinity. Formulate the dual, plot the feasible region of the dual, and show that it is empty. Convert the following optimization problem into a symmetric form linear program, and then find the dual.

4.1.1 Sensitivity Analysis

Suppose we wish to increase profit by expanding manufacturing capacities. In such a situation, it is useful to think of profit as a function of a vector ∆ ∈ <4 of changes to capacity. We denote this profit by z(∆), defined to be the maximal objective value associated with the linear program maximize cT x subject to web.stanford.edu

4.1.3 The Dual Linear Program

Shadow prices solve another linear program, called the dual. In order to distinguish it from the dual, the original linear program of interest – in this case, the one involving decisions on quantities of cars and trucks to build in order to maximize profit – is called the primal. We now formulate the dual. To understand the dual, consider a situati

4.2 Duality Theory

In this section, we develop weak and strong duality in general mathematical terms. This development involves intriguing geometric arguments. Devel-oping intuition about the geometry of duality is often helpful in generating useful insights about optimization problem. Duality theory applies to general linear programs, that can involve greater-than,

4.2.3 First Order Necessary Conditions

It is possible to establish the Strong Duality Theorem directly, but the KKT conditions (given later in this section) are useful in their own right, and strong duality is an immediate consequence. Before given the KKT conditions, we digress still more, and talk about convex sets and hyperplanes. Given two sets, U and V , we say that a hyper-plane H

4.3 Duals of General Linear Programs

For a linear program that isn’t in the symmetric form we can still construct the dual problem. To do this, you can transform the linear program to sym-metric form, and then construct the dual from that. Alternatively, you can apply the KKT conditions directly. Either approach results in an equivalent problem. For example, suppose the linear program

4.4 Two-Player Zero-Sum Games

In this section, we consider games in which each of two opponents selects a strategy and receives a payoff contingent on both his own and his opponent’s selection. We restrict attention here to zero-sum games – those in which a payoff to one player is a loss to his opponent. Let us recall our example from Chapter ?? that illustrates the nature of s

4.5 Allocation of a Labor Force

Our economy presents a network of interdependent industries. Each both produces and consumes goods. For example, the steel industry consumes coal to manufacture steel. Reciprocally, the coal industry requires steel to support its own production processes. Further, each industry may be served by multiple manufacturing technologies, each of which req

M technologies.

In the remainder of this section, we will leverage linear algebra and duality theory to prove the substitution theorem. web.stanford.edu

4.5.1 Labor Minimization

Consider a related problem with an objective of minimizing the labor required to generate a particular “bill of goods” b ∈ web.stanford.edu

4.5.2 Productivity Implies Flexibility

Consider M technologies, each of which produces one of the M goods. To-gether they can be described by an M × M production matrix A. Inter-estingly, if A is productive then any bill of goods can be met exactly by appropriate application of these technologies. This represents a sort of flexi-bility – any demands for goods can be met without any exce

Aˆx < αAx + Aˆx = A(αx + ˆx),

contradicting the fact that at least one component of A(αx+ˆx) is nonpositive. web.stanford.edu

Question 6

Consider a symmetric form primal linear program: maximize cT x subject to Ax ≤ b ≥ 0. Find the dual of this problem. Noting that max f(x) = min −f(x) rearrange the dual into symmetric form. Take the dual of you answer, and rearrange again to get into symmetric form. Explain why the sensitivities of the optimal dual objective to the dual constraints

M = , q = , z = . −AT 0 −b y

Derive the dual Show that the optimal solution to the dual is given by that of the primal and vice-versa. Show that the primal problem has an optimal solution if and only if it is feasible. web.stanford.edu

Question 10

Why is it that, if the primal has unique optimal solution x∗, there is a sufficiently small amount by which c can be altered without changing the optimal solution? web.stanford.edu

Question 13 - Elementary Asset Pricing and Arbitrage

You are examining a market to see if you can find arbitrage opportunities. For the same of simplicity, imagine that there are M states that the market could be in next year, and you are only considering buying a portfolio now, and selling it in a years time. There are also N assets in the market, with price vector ρ ∈ web.stanford.edu

PDF	Lecture 10: Duality and Post-Optimal Analysis Lecture 10: Duality and Post-Optimal Analysis. September 21 2009 of a dual constraint

PDF	Ch.4 Duality and Post Optimal Analysis provides the optimal solution to the other. • The primal problem represents a resource allocation case where the dual problem.

PDF	Linear Programming with Post-Optimality Analyses The original maximization equation/problem is known as the Primal. To calculate the Dual problem the number of variables is calculated by looking at the number

PDF	Lecture 11: Post-Optimal Analysis Sep 23 2009 Dual Simplex Method. • The dual simplex method will be crucial in the post-optimal analysis. • It used when at the current basic solution

PDF	LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS Sensitivity is a post-optimality analysis of a linear program in which some components of (A

PDF	1 Post-optimal analysis This set of notes discusses post-optimal analysis or how to draw conclusions However LINDO also lists values for so-called reduced costs" and dual.

PDF	Lecture 12: Post-Optimal Analysis Sep 25 2009 Then a dual simplex method can be used to determine a new feasible solution. We want x7 to leave (since it is not feasible). Which variable will ...

PDF	Duality in Linear Programming In the preceding chapter on sensitivity analysis we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept.

PDF	Duality in Linear Programming Furthermore an optimal dual variable is nonzero only if its associated constraint In Chapter 3

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PDF	Lecture 10: Duality and Post-Optimal Analysis Duality and Post-Optimal Analysis September 21, 2009 The original LP is referred to as primal • The dual is another LP associated with the given LP

PDF	5 DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY POST-OPTIMALITY 5 1 DUALITY Associated with every linear programming problem (the primal) is Various types of sensitivity analysis can answer such

PDF	Linear Programming with Post-Optimality Analyses This is what essentially all LP packages do in a more efficient way call the Simplex Method Dual Problem and Its Meaning The Dual Analysis involves looking at

PDF	LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS - NC State ISE Sensitivity analysis 4 Dual simplex If x is a primal feasible solution to (P) and w is a dual feasible Sensitivity is a post-optimality analysis of a linear program

PDF	Lesson 5 Slides-Revised Simplex Method, Duality and - NPTEL Primal-Dual relationship is also helpful in sensitivity or post optimality analysis of decision variables Page 3 D Nagesh Kumar, IISc Optimization Methods: M3L5

PDF	Chapter 4 Duality analysis Consider a situation where the metal stamping and engine assembly ca - pacity constraints are binding at the optimal solution to the linear program

PDF	1 Post-optimal analysis A constraint's dual price is also a measure of the decrease in the optimal solution value for each allowable unit of decrease in the RHS value of that constraint

PDF	SENSITIVITY AND DUALITY ANALYSES OF AN OPTIMAL WATER one Operations Researcher to assist them in some of these post-optimality analyses KEYWORDS: Sensitivity Analysis, Duality Analysis, Linear Programming,