How to solve a linear programming problem using the dual simplex method duration. Subramani1 1lane department of computer science and electrical engineering west virginia university february 10, 2015 linear programming optimization methods in finance. 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. The primal and the dual relate to each other in the same way as ows and cuts. Lecture 6 1 the dual of linear program stanford cs theory. Convex optimization, discrete optimization, duality, linear programming. 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. The original linear programming problem is called primal, while the. May 08, 2017 duality 7 duality of an lpp with equality. Examples include the transportation simplex method, the hungarian algorithm for the assignment problem, and the network simplex method. The feasible region of the linear programming problem is empty. These two lp problems are said to be duals of each other.
We will see how to interpret the meanings of the dual decision variables in the context of the original problem, and we will. Duality in generalized linear fractional programming. 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 s0 i, x0 j are a rearrangement of s i, x j while the y0 i, t0 j are the. Linear programming duality part 2 tim roughgardeny february 2, 2016 1 recap this is our third lecture on linear programming, and the second on linear programming duality.
Linear programming notes vi duality and complementary. Understanding the dual problem leads to specialized algorithms for some important classes of linear programming problems. Corresponding to every linear programming problem, there is another linear programming problem. Introduction one of the most important discoveries in the early development of linear programming was the concept of duality and its ramifications. In this article we will discuss about linear programming lp.
Weak duality theorem if lp 1 is a linear program in maximization standard form, lp 2 is a linear program in. We also show a relationship between this result and classical lagrangian duality. There are obvious generalizations that combine these two methods. This paper will cover the main concepts in linear programming, including examples when appropriate. Linear programming duality example linear programming. This can help managers in answering questions about alternative courses of action and their effect on values of the. Duality every linear maximization problem the primal has a dual minimization problem. We will now discuss how to find solutions to a linear programming problem. The original problem is called primal programme and the corresponding unique problem is called dual programme. Lp ii, fall 20 duality page 63 duality theory for linear programming. Strong duality of linear programming hao huang for an m nmatrix a, a vector c2rn and another vector b2rm. That is, if you have one linear programming problem, then there is automatically another one, derived from the same data.
The two programmes are very closely related and optimal solution of. Linear programming, lagrange multipliers, and duality geoff gordon lp. 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. Duality theorem if the primal problem has an optimum then the dual does. Duality in linear programming is essentially a unifying theory that develops the. In which we introduce the theory of duality in linear programming. Combining this result with weak duality, we get zb fvk b. Linear programming notes vi duality and complementary slackness.
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, the original linear program, thereby taking advantage of some computational efficiencies. Applying strong duality theorem in linear programming, we obtain that the dual problem is also solv able and the objectiv e functions of both problems are equal, that is. The given problem is called the primal and the other its dual. Duality in linear programming problems your article library. Linear programming princeton university computer science.
In this chapter, we will develop an understanding of the dual linear program. In this chapter, we will be concerned only with the graphical method. The theory of duality for linear programs is welldeveloped and has been successful in advancing both. Method to solve linear programming maximization problems, em 8720, or another of the sources listed on page 35 for information about the simplex method. Linear programming, lagrange multipliers, and duality. Duality in linear programs 103 by construction, the objective function a b, and we will see in future, in well conditioned problems, the equality of the two objective functions is always observed, i. To this end, we present the theory of duality which provide ways of finding good. In the case of linear programming, duality yields many more amazing results. Solving the linear program associated with an allinteger program gives the group problem n x.
Linear programming is a special case of mathematical programming also known as mathematical optimization. Duality for mixedinteger linear programs optimization online. A comparison of mixedinteger programming models for non. Standard form means, all the variables in the problem should be nonnegative and. An optimum solution is x0 j 0, s 0 i b i, y0i 0 and t j c0 j. This section presents the weak duality theorem in linear programming problems, which. 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. If any of the two problems has an infeasible solution, then the value of the objective. Linear programming duality handout3 may 3, 2015 lpduality asserts that if the optimal solution of the primal linear program below exists and is bounded, then the same holds for the dual linear program, and both optimal solutions. A linear programming model for the transportation problem is then used to show how a class of primal models gives rise to a certain class of dual models. By taking advantage of our knowledge of lp duality, it has been.
Combining all inequalities, we obtain the following minimization linear programme. As illustrations of particular duality rules, we use one small linear program made up for the purpose, and one from a game theory application that we previously developed. Strong duality of linear programming emory university. Chapter 4 duality given any linear program, there is another related linear program called the dual. Lp is a mathematical technique for the analysis of optimum decisions subject to certain constraints in the form of linear inequalities. I the value of any feasible solution to the primal is a lower bound for the. Geometric duality the idea of duality for cones is almost the same as the standard idea of geometric duality. I weak duality i separating hyperplane theorem and theorems of the alternatives i strong duality and complementary slackness i using duality theory to i characterize unbounded lps i resolution theorem and its converse. If we solve this linear program by the simplex method, the resulting optimal solution is y1.
Linear programming duality 1 examples of problems that can be. 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. March 31, 2007 mathematics department brown university 151 thayer street providence, ri 02912 abstract we describe linear programming, an important generalization of linear algebra. Since the problem d is a linear program, it too has a dual.
Merge two adjacent period nodes if the same set of tasks can be scheduled in both periods. In todays lecture we will formally introduce duality and relate it to the toll congestion and maximum weight matching problems from the previous lectures. Ralphs july, 2010 1 introduction this article describes what is known about duality for integer programs. One type of linear program has the form max xn j1 c jx j subject. Fractional programming, quasiconvex programming, duality. However in general the optimal values of the primal and dual problems need not be equal. 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.
Vanderbei october 17, 2007 operations research and financial engineering princeton university princeton, nj 08544. Saltzman march 11, 1998 in this essay, we will \discover the dual problem associated with an lp. This understanding translates to important insights about many optimization problems and algorithms. For this minmax problem a dual in the form of a maxmin problem is introduced and duality relations are established. Duality in mathematical programming lixpolytechnique.
Applications linear programming ii 3 lp duality goal. Balinski and others published duality theory of linear programs. However, the special structure of the transportation problem allows us to solve it with a faster, more economical algorithm than simplex. This lecture begins our discussion of linear programming duality.
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. Weak duality dlp provides upper bound in the case of maximization to the solution of the plp. Furthermore, for the problem considered, the optimal solution helps to infer that by incorporating fuzziness in a linear programming model either in constraints, or both in objective functions and. Also, when solving the dual of any problem, one simultaneously solves the primal. We have seen in lecture 20 that every linear program can be put into canonical form as. Thus, duality is an alternative way of solving lp problems.
The optimal w for the dual problem is the optimal z for the primal problem, both being v in the optimal primal dictionary. Jun 23, 2012 duality is a concept from mathematical programming. Although the idea of duality is essentially mathematical, it has important interpretations. If either the primal or dual problem has a solution then the other also has a solution and their optimum values are equal. One day in 1990, i visited the computer science department of the university of minnesota and met a young graduate student, farid alizadeh. The concepts of both duality and fuzzy uncertainty in linear programming have been theoretically analyzed and comprehensively and practically applied in an abundance of cases. Linear programming has many practical applications in transportation, production planning.
The solution to the dual problem provides a lower bound to the solution of the primal minimization problem. Many practical problems in operations research can be expressed as linear programming problems. The duality theory in linear programming yields plenty of extraordinary results, because of the specific structure of linear programs. In this course, we introduce the basic concepts of linear programming. Linear programming notes vi duality and complementary slackness 1 introduction it turns out that linear programming problems come in pairs. The dual lp is a minimization lp, where the constraints are of. Consider the linear programming problem in standard form. Linear programming is used to successfully model numerous real world situations, ranging. It is perhaps surprising that many of the results familiar from linear programming lp duality do extend to integer programming. Linear programming and duality a reminder of some linear programming vocabulary. Before solving for the duality, the original linear programming problem is to be formulated in its standard form. 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. The duality terminology suggests that the problems p and d come as a pair implying that the dual to d should be. Then we will look at the concept of duality and weak and strong duality theorems.
A constructive approach with applications find, read and cite all. Notes on linear programming james aspnes april 4, 2004 1 linear programming linear programs are a class of combinatorial optimization problems involving minimizing or maximizing a linear function of a of some realvalued variables subject to constraints that are. Knowledge of duality allows one to develop increased insight into lp solution interpretation. Duality in linear programming standard form for linear. We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming lp relaxation of three textbook mixedinteger programmingformulations each approximates the cost function by its lower convex envelope. Lets page back in the relevant stu from last lecture. Choudhary and sinha we begin by looking at some problems that can be cast as linear programming problems.
For every linear programming problem, there is a corresponding unique problem involving the same data and it also describes the original problem. Theorem 1 weak duality if x is a feasible solution to the primal lp 4. Assuming we delete the arrays created after we are done using it, we still use on space for keeping copies of the input array. But first, we need to learn a little bit about linear programming duality. Lecture 7 1 linear programming duality linear programming duality underlies much of what we have been doing in class so far. Maximize ctx subject to ax b primal x 0 its dual linear program is.