Linear programming duality and algorithms lecturer. Duality in linear programming in quantitative techniques for. Ramik 2005 discussed a class of fuzzy linear programming problems based on fuzzy relations and a new concept of duality and deduced the weak and strong duality theorems. Duality in linear programming companion slides of applied mathematical programming by bradley, hax, and magnanti addisonwesley, 1977 prepared by. Lp duality and the primaldual method in the previous few lectures we have seen examples of lprounding, a method for obtaining approximation algorithms that involves solving a linear programming relaxation of the problem at hand and rounding the solution. The constraints, feasible region, and optimal solution of the linear program associated with building cars and trucks. Duality in linear programming in quantitative techniques. Lecture 6 1 the dual of linear program stanford cs theory. Knowledge of duality allows one to develop increased insight into lp solution interpretation. Second, it is often possible to solve the related linear program with the shadow pricesasthevariablesinplaceof,orinconjunctionwith,theoriginallinearprogram,therebytakingadvantage of some computational ef. Linear programming notes vi duality and complementary slackness. 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. For every linear programming problem, there is a corresponding unique problem involving the same data and it also describes the original problem.
Overview this is a tutorial about some interesting math and geometry connected with constrained optimization. If primal has an optimal solution, then there is no duality gap. The linear programming dual of the last problem is the problem 11, 12. In the chapter on sensitivity analysis, we saw that the shadowprice interpretation of the optimal simplex multipliers is a very useful concept. Linear programming is a special case of mathematical programming also known as mathematical optimization more formally, linear programming. In mathematics, linear programming is a method of optimising operations with some constraints. Linear programming has many practical applications in transportation, production planning. Change of variables and normalise the sign of independent terms. This section presents the weak duality theorem in linear programming problems, which claims that the optimal solution to the dual problem. The reduced costs for all nonbasic variables are negative. In solving the primal problem, we have also found a solution to the dual problem. In this way we get the following dual linear program for y1,y2. 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. It turns out that linear programming problems come in pairs.
This procedure, called the simplex method, proceeds by moving from one feasible solution to another, at each step improving the value of the objective function. This understanding translates to important insights about many optimization problems and algorithms. The duality terminology suggests that the problems p and d come as a pair implying that the dual to d should be. Lecture 6 in which we introduce the theory of duality in linear programming. However in general the optimal values of the primal and dual.
Linear programming applications of linear programming. First, these shadow prices give us directly the marginal worth of an additional unit of any of the resources. If the primal problem has an optimal solution, x x 1. Duality in linear programming is essentially a unifying theory that develops the. Thus, duality is an alternative way of solving lp problems. The importance of duality for computational procedures. That is, if you have one linear programming problem, then there is automatically another one, derived from the same data. L ecture linear programming duality 1 introduction consider the following linear program lp 1 in standard form. It is one of the most widely used operations research tools and has been a decisionmaking aid in almost all manufacturing industries and in financial and service organizations. 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. We discuss generalizations to binary integer linear programming with an example of a manager of an activity hall, and conclude with an analysis of versatility of linear programming and the types of. Strong duality of linear programming emory university. A pair of primal and dual linear programs written in the standard form is.
The transpose of a matrix a is found by exchanging the rows and columns. We will now discuss how to find solutions to a linear programming problem. Linear programming notes vi duality and complementary slackness 1 introduction it turns out that linear programming problems come in pairs. It involves slack variables, tableau and pivot variables for the optimisation of a problem. The importance of duality for computational procedures will become. That is, for any production activity at positive level, marginal revenue must equal marginal cost. Duality in fuzzy linear programming with symmetric. Linear programming halfspace,andthereforeanypolyhedron,isconvexifapolyhedroncontainstwopoints x and y,thenitcontainstheentirelinesegmentxy. Chapter 4 duality given any linear program, there is another related linear program called the dual. The dual model is derived by construction from the standard inequality form of linear programming model as shown in tables 1 and 2.
Pdf on aug 10, 2018, dr and others published duality in linear programming find, read and cite all the research you need on researchgate. Solving linear programs 2 in this chapter, we present a systematic procedure for solving linear programs. Neither constraint is tangent to the level surface. Duality in linear programming problems your article library. Pdf on aug 10, 2018, dr and others published duality in linear programming find, read and cite all the research you need on. Also, when solving the dual of any problem, one simultaneously solves the primal. Linear programming, or lp, is a method of allocating resources in an optimal way. Second, when an activity is priced out using these shadow prices, the. The study of duality theory for fuzzy parameter linear programming problems has attracted researchers in fuzzy decision theory. Subramani1 1lane department of computer science and electrical engineering west virginia university february 10, 2015 linear programming optimization methods in finance. Vanderbei october 17, 2007 operations research and financial engineering princeton university princeton, nj 08544. Linear programming, lagrange multipliers, and duality.
So i decided to make todays lecture the one on linear programming and duality, which id planned for friday, and give myself two more days to learn about illposed and inverse problems, and then come back to that friday, so that weve studied the limits in those problems of alpha going to infinity or 0, but the scientific question when theres. Ganesan 2006 established the concept of duality in fuzzy linear programming. The solution to the dual problem provides a lower bound to the solution of the primal minimization problem. Jun 23, 2012 duality is a concept from mathematical programming. Geometric duality the idea of duality for cones is almost the same as the standard idea of.
We describe the types of problems linear programming can handle and show how we can solve them using the simplex method. Because it is often possible to solve the related linear program with the shadow prices as the variables in place of, or in conjunction with. Since the problem d is a linear program, it too has a dual. 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. We will also present several algorithms for solving linear programs.
The original problem is called primal programme and the corresponding unique problem is called dual programme. If we solve this linear program by the simplex method, the resulting optimal. In the case of linear programming, duality yields many more amazing results. Theorem 4 strong duality if x is an optimal primal solution and y is an. The talk is organized around three increasingly sophisticated versions of the lagrange multiplier theorem. Linear programming 507 given sum by the dealer in purchasing chairs and tables is an example of an optimisation problem as well as of a linear programming problem. In the term linear programming, programming refers to mathematical programming. In todays lecture we will formally introduce duality and relate it to the toll congestion and maximum weight matching problems from the previous lectures. Standard form means, all the variables in the problem should be nonnegative and. Michel goemans 1 basics linear programming deals with the problem of optimizing a linear objective function subject to linear equality and inequality constraints on the decision variables. To solve linear programming models, the simplex method is used to find the optimal solution to a problem. In this chapter, we will develop an understanding of the dual linear program.
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. It is the topic of this lecture, the next lecture, and as will become clear pretty much all of the succeeding lectures as well. The values imputed to the resources are such that the net marginal revenue is zero on those activities operated at a positive level. The two programmes are very closely related and optimal solution of. Lecture 5 1 introduction 2 linear programming duality theoretical. Linear programming duality part 1 tim roughgardeny january 28, 2016 1 warmup this lecture begins our discussion of linear programming duality, which is the really the heart and soul of cs261. Strong duality of linear programming hao huang for an m nmatrix a, a vector c2rn and another vector b2rm. Given a feasible solution x of lp and a feasible solution y,s of ld, the duality gap is simply c x. Consider the linear programming problem in standard form. In order to illustrate some applicationsof linear programming,we will explain simpli ed \realworld examples in section 2. Given any linear program, there is another related linear program called the dual.
Maximize ctx subject to ax b primal x 0 its dual linear program is. Linear programming is a special case of mathematical programming also known as mathematical optimization. Then the claim follows directly from duality theorem in linear programming. Linear programming notes vi duality and complementary. The feasible region of the linear programming problem is empty. The duality of fuzzy parameter linear programming was first studied by rodder and zimmermann 1980. Before solving for the duality, the original linear programming problem is to be formulated in its standard form. The main objective of linear programming is to maximize or minimize the numerical value. Strong duality theorem conclusion on previous slide is the essence of the strong duality theorem which we now state. 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. Linear programming lp, also called linear optimization is a method to achieve the best outcome such as maximum profit or lowest cost in a mathematical model whose requirements are represented by linear relationships. We will write the standard linear programming dual problem as. This paper will cover the main concepts in linear programming, including examples when appropriate. Tianqi song 1 overview in this lecture, we will cover more examples of linear programming and introduce linear programming duality.
Solve the following linear program using the primal simplex. Oct 19, 2016 29 videos play all math484, linear programming, fall 2016 wenshenpsu the most beautiful equation in math duration. In discussing duality, it is common to depart from the standard equality form of the lp given in section 4. It consists of linear functions which are subjected to the constraints in the form of linear equations or in the form of inequalities linear programming is widely used in mathematics and some. Linear programming, lagrange multipliers, and duality geoff gordon lp. Tianqi song 1 overview in this lecture, we will cover more examples of linear programming and introduce linear program ming duality. We will end with a study of the dual of maxflow problem. In this chapter, we will be concerned only with the graphical method. Linear programming duality 1 examples of problems that can be. The duality theory in linear programming yields plenty of extraordinary results, because of the specific structure of linear programs.
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