If either the primal of dual problem has a solution, then the other also has a solution and their optimum values are equal. Duality in linear programming is essentially a unifying theory that develops the. The study of such changes is called sensitivity analysis. Theres something else quite interesting about duality.
Lecture 6 1 the dual of linear program stanford cs theory. It has been used successfully as a decisionmaking aid in. Sensitivity analysis allows him to ask certain whatif questions about the problem. Therefore it is sufficient to solve one of them primal or dual to obtain the optimal solution and the optimal value of the equivalent problem primal or. I strong duality and complementary slackness i using duality theory to i characterize unbounded lps i resolution theorem and its converse. Shadow price in a linear programming problem in standard form, shadow price is the amount by which the optimal value of the objective function is improved so that it increases the z value for a maximization problem and decreases the z value for a minimization problem if the right hand side of constraint is increased by one unit and all other. The duality in linear programming states that every linear programming problem has another linear programming problem related to it and thus can be derived from it. Slaters conditions holds if the primal is feasible, i.
Duality in linear programming is essentially a unifying theory that develops the relationships between a given linear program and another related linear program stated in terms of variables with this shadowprice interpretation. Characteristics of dual problem, advantages of duality. Mike spiveys blog post gives some interesting examples of some nice properties of duality but i was wondering if there are algorithmic advances, for example, that come from understanding duality or anything more cs related. It is one of the most widely used operations research or tools. Duality in linear programming problems your article library. Any linear programming problem marked as p and called primal can be seen in connection with another linear programming problem marked as d and called dual. The duality theory in linear programming yields plenty of extraordinary results, because of the specific structure of linear programs. Maximize ctx subject to ax b primal x 0 its dual linear program is. This understanding translates to important insights about many optimization problems and algorithms. Weak and strong duality geometric interpretation saddlepoint interpretation optimality conditions perturbation and sensitivity analysis nonlinear optimization c 2006 jeanphilippe vert, jeanphilippe. There is a software called gipels available on the internet which easily solves the lpp problems along with the transportation problems.
The dual in linear programming in lp the solution for the profitmaximizing combination of outputs automatically determines the input amounts that must be used in the production process. Then the claim follows directly from duality theorem in linear programming. Thus, duality is an alternative way of solving lp problems. Minimize bty subject to aty c dual y 0 the weak duality says that if x 0 is a feasible solution to the primal, and y.
Duality is a unifying theory that develops the relationships between a given linear program and another related linear program stated in terms of variables with this shadowprice interpretation. Examples include the transportation simplex method, the hungarian algorithm for the assignment problem, and the network simplex method. The economic interpretation of the dual model brings about new information when analyzing such phenomena and when substantiating decision making. In the primal problem, the objective function is a linear combination of n variables. Does the duality theorem of linear programming hold only in closed convex cones.
Strong duality of linear programming hao huang for an m nmatrix a, a vector c2rn and another vector b2rm. It provides a refreshing and thoroughly unique perspective on linear programming that is imbued with pariss characteristic passion for economics in general, and duality and symmetry in particular beautiful is the word that. Here is the video about linear programming problem lpp using dual simplex method minimization in operations research, in this video we discussed briefly and solved. The idea behind duality lec12p1, orf363cos323 princeton. Given any linear program, there is another related linear program called the. Strong duality of linear programming emory university. I understand the mechanics of solving a dual problem i do not need help with that. Understanding the dual problem leads to specialized algorithms for some important classes of linear programming problems. In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. It has been used successfully as a decisionmaking aid in almost all industries and in financial and service organizations.
What is the significance of duality theory of linear. Geometry of lp duality linear programming duality coursera. It consists in optimizing a linear objective subject to linear constraints, admits efficient algorithmic solutions, and is often an important building block for other optimization techniques. The original linear programming problem is called primal, while the. The linear programming method is a technique of selecting the best alternative out of the available set of feasible alternatives, for which the objective function and the constraint function can be expressed as linear mathematical functions. Lp 6 duality interpretation and uses linear programming coursera. There are m constraints, each of which places an upper bound on a linear combination of the n variables. Geometric interpretation of duality in optimization. Linear programming problems are optimization problems in which the objective function and the constraints are all linear. For every linear programming problem, there is a corresponding unique problem involving the same data and it also describes the original problem. Economic interpretations primal linear problem primal lp problem can be viewed as a resource allocation model that seeks to maximize the revenueunder limited resources primal problem n economic activities m resources.
Duality in linear programming linear programming duality duality theorem. What is the meaning of these dual variables, and then i want to talk about, you. An economic interpretation of linear programming is a serious and meaningful revision of an already excellent book. Linear programming and optimization are used in various industries. Also, when solving the dual of any problem, one simultaneously solves the primal. Lp ii, fall 20 duality page 63 duality theory for linear programming i special case of lagrangian duality theory for general optimization i idea. In fact, the dual of any optimization problem often has a nice interpretation. The dual of a given linear program lp is another lp that is derived from the original the primal lp in the following schematic way. In the case of linear programming, duality yields many more amazing results. The original problem is called primal programme and the corresponding unique problem is called dual programme. The solution to either is sufficient for readily obtaining. I have just learned the simplex method for solving linear programs, and im trying to understand what its dual problem represents.
Sensitivity analysis addresses the question of how optimal solutions depend on perturba. In this chapter, we will develop an understanding of the dual linear program. Economic interpretation of duality free download as powerpoint presentation. Open educational resource using duality and sensitivity. Knowledge of duality allows one to develop increased insight into lp solution interpretation. Sensitivity is a postoptimality analysis of a linear program in which, some components of a, b, c may change after obtaining an optimalsolution with an optimal basis and an optimal objective value. Economic interpretations of linear programming problem 1. First, these shadow prices give us directly the marginal worth of an additional unit of any of the resources. Lp 6 duality interpretation and uses linear programming. Using duality and sensitivity analysis to interpret linear. Since the problem d is a linear program, it too has a dual. Duality in linear programming has the following major characteristics. In the case of linear programming, duality yields many more amazing. We have recently covered linear programming and i am comfortable with the weak and strong duality theorems.
Jan 15, 2015 by linear programming webmaster on january 15, 2015 in linear programming lp the dual model of a linear programming problem consists of an alternative modeling instance that allows us to recover the information of the original problem commonly known as primal model. Before solving for the duality, the original linear programming problem is to be formulated in its standard form. The objective direction is inversed maximum in the primal becomes minimum in the dual and viceversa. Vandenberghe is an associate professor at the department of electrical engineering, university of. Lp ii, fall 20 duality page 63 duality theory for linear programming. Solve the following linear program using the primal simplex.
Linear programming has been, and remains, a workhorse of optimization. What is the physical interpretation of the dual variables or u. If any of the two problems has an infeasible solution, then the value of the objective. If either the primal or dual problem has a solution then the other also has a solution and their optimum values are equal. I understand the mechanics of solving a dual problem. If we solve this linear program by the simplex method, the resulting optimal solution is y1 11. However, given todays computer capabilities, this is an infrequently used aspect of duality. Duality in linear programming duality optimal solution of d y 5 2. The duality terminology suggests that the problems p and d come as a pair implying that the dual to d should be. This lectures notes focus on the duality in linear programming, and give an example of the dual problem for maximum ow problem. A technical explanation of duality that attempts to offer some intuitions including the insight that the primal and dual each sort of encode the other, so that you could reverse engineer one in order to reconstruct the other. Duality in linear programming 4 in the preceding chapter on sensitivity analysis, we saw that the shadowprice interpretation of the optimal simplex multipliers is a very useful concept. If one problem has an optimal solution, than the optimal values are equal. Music let us now look at linear programming duality from a geometry perspective.
Its now time for you to learn about the dual linear program. Lecture 6 in which we introduce the theory of duality in linear programming. Standard form means, all the variables in the problem should be nonnegative and. In this section, we are going to look at the various applications of linear programming. In solving the primal problem, we have also found a solution to the dual problem. However, i dont understand what the applications of duality are that are specific to tc. Of course, once you relax, ooh, this is an interesting linear program, you can start. Sensitivity analysis and interpretation of solution introduction to sensitivity analysis.
The manufacturing and service industry uses linear programming on a regular basis. Linear programming duality theory formulation, solutions and interpretation vidyamitra. The solution to the dual problem provides a lower bound to the solution of the primal minimization problem. Duality in linear programming is essentially a unifying theory that develops. Economic interpretations of linear programming problem. Geometric interpretation of duality and slaters condition. Lagrangian duality cu denver optimization student wiki. By linear programming webmaster on january 15, 2015 in linear programming lp the dual model of a linear programming problem consists of an alternative modeling instance that allows us to recover the information of the original problem commonly known as primal model. Open educational resource using duality and sensitivity analysis to interpret linear programming solutions public deposited.
However, i dont understand what the applications of duality are that are specific to tcs. The two programmes are very closely related and optimal solution of. The dual model of a linear programming problem consists of an alternative modeling instance that allows us to recover the information of the original problem commonly known as primal model. Here you will learn linear programming duality applied to the design of some approximation algorithms, and semidefinite programming applied to maxcut. The problem p has an optimal solution if and only if the dual problem d has an optimal solution. The weak duality theorem states that the objective value of the dual lp at any feasible solution is.
The original linear programming problem is called primal, while the derived linear problem is called dual. For any linear program lp, there is a closely related lp called the dual. Economic interpretation of duality, shadow price and complementary slackness theorem. Linear programming simplex algorithm, duality and dual simplex algorithm martin branda charles university in prague faculty of mathematics and physics department of probability and mathematical statistics computational aspects of optimization 1 32. Linear programming, or lp, is a method of allocating resources in an optimal way. Economic interpretation of duality linear programming.
If the primal involves n variables and m constraints, the dual involves n constraints and m variables. By taking the two parts of this course, you will be exposed to a range of problems at the foundations of theoretical computer science, and to powerful design and analysis techniques. Duality theory applies to general linear programs, that can involve greater. However in general the optimal values of the primal and dual problems need not be equal. Chapter 4 duality given any linear program, there is another related linear program called the dual. Manufacturing industries use linear programming for analyzing their supply chain operations. There are several beautifully written posts on stackexchange about duality. This presentation is trying to explain the linear programming in operations research. Duality topics are our second theoretical unit and after which, we will cover the secondorder algorithms. Using duality and sensitivity analysis to interpret linear programming solutions j. Understanding the lagrangian dual problem for nonlinear programming is the foundation for understanding the theory behind duality in optimization research and the ways that the dual program can be used to find optimal solutions to the primal program. The linear programming dual of the last problem is the problem 11, 12. Duality in linear programming in quantitative techniques. Duality in linear programming in quantitative techniques for.
Jun 23, 2012 duality is a concept from mathematical programming. In this case, the shadow prices are interpreted as the opportunity. Linear programming duality theory formulation, solutions. Duality linear programming mathematical optimization. Linear programming applications of linear programming. Linear programming simplex algorithm, duality and dual. Primal dual relationships in linear programming duality. Linear programming simplex algorithm, duality and dual simplex algorithm martin branda.
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