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Math 407 — Linear Optimization 1 Introduction

Math 407 Linear Optimization1 What is Optimization ?A mathematical Optimization problem is one in which some function is either maximized orminimized relative to a given set of alternatives. The function to be minimized or maximizedis called theobjective functionand the set of alternatives is called the feasible region (orconstraint region). In this course, the feasible region is always taken to be a subset ofRn(realn-dimensional space) and the objective function is a function further restrict the class of Optimization problems that we consider to Linear program-ming problems (or LPs). An LP is an Optimization problem overRnwherein the objectivefunction is a Linear function, that is, the objective has the formc1x1+c2x2+ +cnxnfor someci2Ri=1.

The next step is to draw in the vector representing the gradient of the objective function. This vector may be placed anywhere on your graph, but it is often convenient to draw it emanating from the origin. Since the objective function has the form f(x 1,x 2)=c 1x 1 +c 2x 2, the gradient of f is the same at every point in R2; rf(x 1,x 2)= c 1 c 2 .

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