We study the modeling of nonconvex piecewise linear functions as mixed integer programming mip problems. Continuous piecewise linear functions and their discontinuous extensions are also extensively studied within discrete optimization and mixed integer programming mip, e. Hochbaum contents 1 introduction 1 2 formulation of some ilp 2 2. Page 1 of 2 114 chapter 2 linear equations and functions piecewise functions representing piecewise functions up to now in this chapter a function has been represented by a single equation. A comparison of mixed integer programming models for nonconvex piecewise linear cost minimization problems. Separating we can separate q by factorization and introduction of new variables. This problem is called the linear integerprogramming problem. Piecewise linear approximation of functions of two variables. Pdf a comparison of mixed integer programming models for.
Basic math, prealgebra, beginning algebra, intermediate algebra, advanced algebra, precalculus, trigonometry, and calculus practice problems here. Rn r is convex piecewiselinear if it can be expressed as fx max. Optimization problems involving nonconvex piecewise linear functions can be solved with spe cialized. Piecewiselinear programs several kinds of linear programming problems use functions that are not really linear, but are pieced together from connected linear segments. Piecewise linear approximations do introduce binary integer variables into your model. Linear programming problem feasible region travel salesman problem piecewise linear function integer solution these keywords were added by machine and not by the authors. Steffen rebennack,a, vitaliy kraskob a institute of operations research, karlsruhe. This video shows an application of piecewise linear models and shows the graphs and equations behind these functions. By using 01 variables, however, piecewise linear functions can be represented in linear form. We also remove the inte command which indicates that the variable must have a value of either 0 or 1. By using 01 variables, however, piecewise linear functions. Piecewise linear cost functions we can use binary variables to formulatearbitrary piecewise linear cost functions. A scip constraint handler for piecewise linear functions opus 4. Mathematical programming formulations for piecewise.
Many objective functions, however, tend to be nonlinear. Comparisons of break points selection strategies for. We compare these formulations with respect to their theoretical properties and their relative computational performance. A piecewise function is a function where more than one formula is used to define the output over different pieces of the domain we use piecewise functions to describe situations where a rule or relationship changes as the input value crosses certain boundaries. Piecewise linear approximations for the staticdynamic uncertainty strategy in stochastic lotsizing. A piecewise linear function is not a linear function, so one might think that linear programming could not be used to solve optimization problems involving these functions. For expressing a piecewise linear function of a single variable x with m 1 break. Uwmadison integer programming modeling lecture notes 9 45. Finding all solution sets of piecewiselinear interval. We present novel mixedinteger programming mip formulations for optimization over nonconvex piece wise linear. Second, we consider basic approaches that have been developed for solving integer and mixed integer programming problems. For simplicity, we assume that the function domain d is bounded and f is only used in a constraint of the form fx 0 or as an objective function that is being minimized. Lp relaxation is model of lower convex envelope sharp. That is, fcan be described by a partition of the domain dinto a nite family tciud i 1 of polyhedral pieces, where for each piece c i there is an a ne function fi.
Locally ideal formulations for piecewise linear functions. I do remember that function and something about an integer linear programming. Introduction consider a piecewise linear function f. Modeling variables for pieces sos2, binary formulations for nonconvexity. The above piecewise function itself could be used as a piecewise linear approximation of the nonlinear function as shown below. Examples of valid and nonvalid triangulations in two dimen sions are given.
We also show a relationship between this result and classical lagrangian duality theory. In our case, if our model works only with continuous variables, it will automatically change from being a linear problem to a much more complex integer linear problem. In this paper, we consider mixed integer linear programming mip formulations for piecewise linear functions plfs that are evaluated when an indicator variable is turned on. Mixed integer programming models for nonseparable piecewise linear cost functions juan pablo vielma shabbir ahmed george nemhauser h. Voiceover by now were used to seeing functions defined like hyy2 or fx to the square root of x. Milton stewart school of industrial and systems engineering georgia institute of technology. Hi, mita and i are here to introduce a tutorial on. These models yield optimal continuous pwl functions for a set of discrete data. All models except for cc have integral vertices locally ideal. We can use binary variables to model arbitrary piecewise linear cost functions. Either a mixed integer linear programming problem must be solved as discussed in section 8. Define and write piecewise functions intermediate algebra. Unifying framework and extensions, vielma, ahmed and nemhauser, 2009 38. Mixed integer linear programming formulation techniques.
We consider three easytoimplement methods for the piecewise linear approximation of functions of two variables. If the domain of the function is compact disambiguation needed, there needs to be a finite collection of such intervals. Piecewise linear functions may be defined on ndimensional euclidean space, or more generally any vector space or affine space, as well as on piecewise linear manifolds, simplicial complexes, and so forth. For more information on piecewise linear approximation of functions of two variables see e. However, most related work in this eld concentrates on the modeling of a given piecewise linear function. Models for representing piecewise linear cost functions. Integer programming with piecewise linear functions. Here are the topics that she loves math covers, as expanded below. The points where the slope of the piecewise linear function changes are called the break points of the function. Ima new directions short course on mathematical optimization. Model lowdimensional piecewise linear functions je linderoth uwmadison integer programming modeling lecture notes 2 45. This chapter introduces ampls piecewiselinear notation through examples of.
Piecewise linear approximation of functions of two. Sensitivity analysis with the branchandbound method we input the normal lindo code but remove the gin command which indicates that the variable must be a nonnegative integer value. For a free download of the complete book in pdf format, please visit. Convert a piecewise linear nonc onvex function into a linear optimisation problem. Solving optimization problems containing piecewise linear. Recently various mixed integer programming models for plfs have been proposed by kontogiorgis 3, padberg. In this paper, we propose the first convex model for optimal continuous univariate pwl function fitting.
R we have that epif can be modeled as a binary mixedinteger programming model if and only if fis piecewise linear and lower semicontinuous. Mathematical programming formulations for piecewise polynomial. Hi, mita and i are here to introduce a tutorial on integer programming modeling. Pdf a comparison of mixedinteger programming models for. For nonconvex problems, the approach is still valid but more work needs to be done.
Defines mixed integer linear programming and describes an upgrade to the productionplanning problem to include a fixed charge for the products. The purpose of doing a piecewise linear approximation is that the new linearity will allow the previously nonlinear problem to be solved by linear programming methods, which are much easier to employ than their. Today we start talking about the mixed integer linear program. We adopt disjunctive representations of pwp constraints, drawing upon the extensive work on disjunctive programming dp formulations and representation of piecewise linear pwl functions 2,54,69,76. Toevaluate the function, we take linear combinations p k. In this work, we propose a mixed integer linear programming milp framework for generating optimal piecewise polynomial approximations of varying degrees to nonlinear functions of a single variable. We experimentally evaluate their approximation quality, and give a detailed description of how the methods can be embedded in a milp model. Aimms modeling guide integer programming tricks this. For example, integer variables can be introduced so that a nonlinear function can be approximated by a piecewise linear function. Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programmingbased solution methods. Ieor 269, spring 2010 integer programming and combinatorial optimization professor dorit s. Many objective functions, however, tend to be non linear. Solving optimization problems containing piecewise linear functions joey huchette, mit.
We describe modifications to standard mip formulations for plfs with desirable theoretical properties and superior computational performance in this context. Pdf a comparison of mixed integer programming models. Solving mixedinteger programming problems using piecewise. Piecewise polynomial interpolations and approximations of. Piecewise linear approximations of the standard normal.
This process is experimental and the keywords may be updated as the learning algorithm improves. Pdf mixedinteger models for nonseparable piecewise. The notion of a piecewise linear function makes sense in several different contexts. Mixedinteger models for nonseparable piecewiselinear.
These piecewise linear terms are easy to imagine, but can be hard to describe in conventional algebraic notation. Mixed integer programming models for nonseparable piecewise. A comparison of mixedinteger programming models for non. Mixedinteger models for nonseparable piecewise linear. In this section, we will show how to formulate an integer programming approximation to a general nonlinear term in the objective function. Math3902 operations research ii integer programming p. Branchandbound for biobjective mixed integer programming. Convert a piecewise linear nonconvex function into a. We describe modi cations to standard mip formulations for plfs with desirable theoretical properties and superior computational performance in this context. Piecewise linear programming defines piecewise linear programming, describes an inventory problem with piecewise linear functions, compares pwl to plain linear programming, and indicates complexity issues.
Integer programming modeling ima new directions short course on mathematical optimization je linderoth department of industrial and systems engineering wisconsin institutes of discovery university of wisconsinmadison august 10, 2016 je linderoth uwmadison integer programming modeling lecture notes 1 45. But what were now going to explore is functions that are defined piece by piece over different intervals and functions like this youll sometimes view them as a piecewise, or these types of function definitions they might be called a piecewise function definition. The first step is to approximate the nonlinear function by a continuous piecewise linear function. Recently various mixed integer programming models for plfs have been proposed by kontogiorgis 3, padberg 4, croxton et al. Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programming based solution methods. Formulations with integer variables from a practical standpoint, why do we needinteger variables. Piecewise polynomial functions are extensively used to approximate general nonlinear functions or sets of data. Mass balance or limits on resource use, for example, are generally linear functions. The function is specified by ordered pairs ai,fai and we wish to.
Conn and mongeau 8 constructed a method based on nondifferentiable penalty functions for solving discontinuous piecewise linear optimization problems, sketching an extension to problems with. Global solutions can then be obtained with any number efficient lp codes. Figure 1 a continuous piecewise linear function and its epigraph as the union of polyhedra. In this method, the problem of nding all solution sets is formulated as a mixed integer programming problem, and it is solved by a highperformance in teger programming solver such as cplex. Sensitivity analysis integer programming with piecewise. Mixed integer models for nonseparable piecewise linear optimization. In many reallife problems, however, functions are represented by a combination of. Integer programming modeling institute for mathematics. Convert a piecewise linear nonconvex function into a linear optimisation problem. We have seen in the last lecture that integer variable essentially allow us to introduce disjunctive logic if the variable is associated with a physical entity that isindivisible, then the value must be integer.
Piecewise linear retractions by reflexion pdf free download. A modeling language for mathematical programming varying numbers of pieces 368 17. But with modern lpmip solvers that efficiently handle sets of binary integer variables, you can often find solutions very quickly to problems involving such approximations. How to formulate a piecewise linear function decide. The reflexion method of motzkin and schoenberg is used to generate a class of piecewise linear retractions from x to y w. Nonconvex piecewise linear functions optimization online. Piecewise polynomial interpolations and approximations of one. While this prevents us from including networks that. He showed that the feasible set of the linear programming relaxation of the incremental cost formulation is integral. These piecewiselinear terms are easy to imagine, but can be hard to describe in conventional algebraic notation. Solving mixed integer programming problems using piecewise linearization methods bachelor thesis by bernreuther, marco reg. Padberg compared the linear programming relaxations of the two mixed integer programming models for piecewise linear functions in the simplest case when no constraint exists. We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming lp relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope. Piecewise linear function fitting via mixedinteger linear.
Another type of nonlinear function that can be represented by integer variables is a piecewise linear. So much so, that mixed integer problems can bene t immensely. Introduction to piecewise functions algebra video khan. Piecewise linear function fitting via mixedinteger. Such functions are called for example, the piecewise function given by. Due to the recent developments in commercial mixed integer linear programming milp solvers, it is possible to approximate the nonlinear and nonconvex hpf through piecewise linear pwl models. Piecewise linear bounding of univariate nonlinear functions. In this work, we propose a mixed integer linear programming milp framework for generating optimal piecewise polynomial approximations of varying degrees to nonlinear functions. Piecewise linear pwl function convex and convex relaxation. Integer programming ise 418 lecture 2 lehigh university. Lecture 2 piecewiselinear optimization piecewise linear minimization.
Rn r is convex piecewiselinear if it can be expressed as fx max i1. A piecewise linear function is a function defined on a possibly unbounded interval of real numbers, such that there is a collection of intervals on each of which the function is an affine function. Integer programming formulations mit opencourseware. Dependent on the metrics chosen, the resulting formulations are either mixed integer linear programming or mixed integer quadratic programming problems. Aimms modeling guide integer programming tricks researchgate. Discrete piecewise linear functions on piecewise linear processes more on piecewise linear approximation modified. No general algorithm is known that allows to optimize a solution. Transform functions with products of binary and continuous variables. I do remember that function and something about an integer linear programming problem bclc 11 mins ago.