1 15.Math-Review Wednesday 8/16/00

15.Math-Review2  We can use the derivatives of a function to obtain information about the maximum and minimum values the function attains. zWe saw this for: yfunctions of one variable yunconstrained variables yconstrained variables. Here we have to be careful with the extreme points of the domain. zWe try to locate the local minima/maxima and select the global optimum from them. Optimization

15.Math-Review3 zUnconstrained optimization yGiven f(x), find f ’(x) and f ”(x). ySolve for x such that f ’(x) = 0. ySubstitute the solution(s) into f ”(x). xIf f ”(x)  0, x is a local minimum. xIf f ”(x)  0, x is a local maximum. xIf f ”(x) = 0, x is likely a point of inflection. Optimization

15.Math-Review4 zExample: Due to the interaction of supply and demand, we are able to affect p the price of door knobs with the quantity q of door knobs produced according to the following linear model: p = q zFind the production level that maximizes the revenue. zIs this an example of constrained or unconstrained optimization? Optimization

15.Math-Review5 zConstrained optimization zFunction f(x) defined for x  [a, b] ySame analysis as unconstrained optimization in the interior of the domain, i.e. x  (a, b). yf’(a)  0, a is a local minimum yf’(a)  0, a is a local maximum yf’(b)  0, b is a local maximum yf’(b)  0, b is a local minimum Optimization

15.Math-Review6 zConstrained optimization yIn the following example ypoints a 2, a 3 are obtained by the interior analysis. yPoint a 1 is a local maximum since f’(a 1 )<0 and the domain starts at a 1. ypoint a 4 is a local minimum since f’(a 4 )<0 and the domain ends at a 4. Optimization a1a1 a2a2 a3a3 a4a4

15.Math-Review7 zExample: In the same example of the linear model that related price with production level p = q zConsider now variable operative costs = 20q zMaximize profit, with the consideration that the production level has to be higher than 450 units due to contracts with clients. Optimization

15.Math-Review8 zFor unconstrained minimization of a multivariate function, the previous scheme generalizes easily. y1st order condition: y2nd order condition: yIf these two conditions hold, x is a local minimum. 1st and 2nd order conditions

15.Math-Review9 zFor unconstrained minimization of a multivariate function, the previous scheme generalizes easily. y1st order condition: y2nd order condition: yIf these two conditions hold, x is a local maximum. 1st and 2nd order conditions

15.Math-Review10 zLinear optimization problems consists of selecting the variables that optimize a linear objective function subject to linear constraints. zFor example: Linear Optimization

15.Math-Review11 zThe constraints define the feasible region, in our example, we have: Linear Optimization x4x4 4x+y  20 3x+4y  36

15.Math-Review12 zNote that the feasible region is bounded by lines and has vertices. zWe will always find an optimal solution among the vertices of the polygon that is the feasible region. zThe simplex algorithm uses a clever scheme to search for the optimal among the vertices. zThe feasible region can also be unbounded (polytope) or even not exist. Linear Optimization

15.Math-Review13 zWe can solve small examples graphically. zFor this we have to see the direction of the objective function and the shape of the feasible region. xAdd the level sets of the objective function, i.e. 5x+7y=k, for different values of k. Linear Optimization k=0.0 k=20.0 k=62.15 k=63.0

15.Math-Review14 zExercise: A furniture manufacturer has exactly 260 pounds of plastic and 240 pounds of wood available each week for the production of two products: X and Y. Each unit of X produced requires 20 pounds of plastic and 15 pounds of wood. Each unit of Y requires 10 pounds of plastic and 12 pounds of wood. If the prices for products X and Y are both equal to $10, what is the production level that maximizes the income? zLook at the feasible region, solve graphically. The Linear Equation II

15.Math-Review15 zWhat happens if we tilt the objective function? yMultiple optima. yChange optimal solution from one vertex to the next. Linear Optimization 6x+8y=k 5x+y=k

15.Math-Review16 zIssues in Linear Optimization yUnbounded feasible region, and unbounded optimization problem ySensitivity analysis of optimal solution (dual variables=prices) yDual problem yWeak and strong duality yHow do you go about solving these in practice. Well solved problem. (>1M variables are tackleable) Linear Optimization

15.Math-Review17 zExercise: Graphically solve the following LP. Linear Optimization

15.Math-Review18 zExercise: A computer parts manufacturer produces two types of monitors, monochrome and color. The total daily capacity is 700 and 500 units per day for monochrome and color respectively. For both types, department A has to install the tubes, it takes 1hr for a monochrome and 2 hr for a color. Total daily labor hrs in department A is 1,200 hrs. Department B has to inspect all the monitors, it takes 3 hrs of labor to inspect a monochrome and 2 hrs for a color. Total daily labor hrs in department B is 2,400 hrs. zIf the net contribution to earnings are $40 and $30 per unit, for monochrome and color monitors respectively. How many of each should we produce to maximize the earnings. Linear Optimization

15.Math-Review19 zWe will address problems of the form: zWe construct the Lagrangian function: zSolve the multivariate unconstrained optimization problem for the Lagrangian function: Equality Constrained Optimization

15.Math-Review20 zWe look for x and y that satisfy the 1st order optimality conditions for L(x,y): zunder appropriate conditions on f(x) and g(x) the 1st order conditions of the Lagrangian are enough to guarantee optimality. Equality Constrained Optimization

15.Math-Review21 zExample: A very particular person has an utility function for apples and oranges that follows the expression: U=A 0.4 O 0.6. Apples cost $1.00, and oranges cost $0.50, maximize the utility subject to spending $ zWork with optimizing ln(U) which will optimize U. Equality Constrained Optimization

15.Math-Review22 zExample: Suppose we have the following model to explain q, the quantity of knobs produced: q=L 0.3 K 0.9, where: yL: Labor, and has a cost of $1 per unit of labor. yK: Capital, and has a cost of $2 per unit of capital. zInterpret the model. Is it reasonable? (not the units, please) zFind the mix of labor and capital that will produce q=100 at minimum cost. Equality Constrained Optimization