MatLab Code for Simple Convex Minimization

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1 MatLab Code for Simple Convex Minimization, 1-8, 2017 Draft Version April 25, 2018 MatLab Code for Simple Convex Minimization James K. Peterson 1 * Abstract We look at MatLab code to solve a simple L 1 minimization. Keywords Convex Functions Subgradients MatLab Code 1 Department of Mathematical Sciences, Clemson University,Clemson, SC: petersj@clemson.edu *Corresponding author: petersj@clemson.edu 1. Introduction We are going to solve the cooling model project using ideas from convex analysis. Given data X,Y ) which consists of time and temperature pairs for a cooling liquid, we can fit a Newton cooling model to the data using a Least Squares minimization approach. We assume the temperature T is determined by a cooling model T (t) = A + (T 0 A)e kt. (1) where A is the room temperature where the data pairs have been collected. Let the variable U(t) be defined to be ( ) T (t) A U(t) = ln T 0 A Then, from Equation 1 we have U(t) = kt which is a linear relationship. From the data pairs, this means we want to find a value of k that gives model values A + (T 0 A)e kt i which are close to the measured temperature T i for all the data points. This is equivalent to finding a value of k so that U(t i ) = kt i is close to the value U(t i ) = T i A T 0 A for the data pairs. Let s assume there are N data pairs. Setting up a least squares error function LS we define E LS = N (A + (T 0 A)e kt i T i ) 2 i=1 and as discussed in Project Two, the optimal value of k is the one from the standard regression fit. The details of this derivation are in the writeup for Project Two. However, another way to find an optimal value of k is to use an L 1 error function E One defined by E One = N A + (T 0 A)e kt i T i i=1

2 MatLab Code for Simple Convex Minimization 2/8 Of course, we don t solve these problems as stated. Instead, we follow the ideas in the cooling project to define the auxiliary variable U and set up the errors: E One = E LS = N U i mt i i=1 N (U i mt i ) 2 i=1 Now let s write MatLab code to solve this. The least squares part is like before but the summed absolute error part uses the subdifferentials which is different. 2. The MatLab Implementation of the E One Minimization The MatLab code to solve the E LS minimization problem is discussed in the Project Two handout. Here we will show you the MatLab code to solve the minimization problem for L One which requires a subgradient approach. We know absolute minimum of the convex function E One occurs at an interior point p where 0 E One (p). Here is the code to find this point. Listing 1. LOne Minimization 1 f u n c t i o n [ mstar, E, Df, LSmstar, LSE ] = LOneEnergy (X,Y) % X = x d a t a % Y = y d a t a % mstar = o p t i m a l s l o p e f o r L One % E = o p t i m a l L One v a l u e 6 % LSmstar = o p t i m a l s l o p e f o r L Two % LSE = o p t i m a l L Two v a l u e % Df = s u b g r a d i e n t s a t t h e k n o t s % We assume t h e f i r s t X v a l u e i s z e r o % ( 0,Y( 1 ) ) d o e s n o t e f f e c t t h e e n e r g y f u n c t i o n s 11 [ n,m] = s i z e (X) ; X1 = X( 2 : n ) ; Y1 = Y( 2 : n ) ; Q = abs (Y1. / X1) ; W = s o r t (Z) ; 16 Z = X1. W; Energy Y1,m) sum ( abs (X1). abs (m Y1. / X1) ) ; Df = z e r o s ( n 1,2) ; [ n,m] = s i z e (X) ; f o r i = 1 : n 1 21 belowknot = 0. 0 ; aboveknot = 0. 0 ; f o r j = 1 : n 1 i f W( j ) < W( i ) belowknot = belowknot + X1( j ) ; 26 e l s e aboveknot = aboveknot X1( j ) ; c = belowknot+aboveknot ;

3 MatLab Code for Simple Convex Minimization 3/8 Df ( i, 1 ) = X1( i ) + c ; 31 Df ( i, 2 ) = X1( i ) + c ; mstar = 0. 0 ; f o r i = 1 : n 1 36 i f ( Df ( i, 1 ) < 0 && 0 < Df ( i, 2 ) ) % p r i n t o u t some d i a g n o s t i c s d i s p l a y ( i ) ; d i s p l a y ( Df ( i, 1 ) ) ; d i s p l a y ( Df ( i, 2 ) ) ; 41 mstar = Y1( i ) /X1( i ) ; break ; E = Energy (X1, Y1, mstar ) ; 46 % Compare t o LS LSEnergy Y1,m) sum ( (Y1 m. X1). ˆ 2 ) ; LSmstar = sum (X1. Y1) /sum (X1. X1) ; LSE = LSEnergy (X1, Y1, LSmstar ) ; f 1=s u b p l o t ( 2, 1, 1 ) ; 51 M = l i n s p a c e ( mstar , mstar , 3 1 ) ; PE = Energy (X1, Y1,M) ; p l o t ( f1,m, PE, o ) ; x l a b e l ( m ) ; y l a b e l ( E n e r g y ) ; 56 t i t l e ( L1 E n e r g y v e r s u s s l o p e ) ; f 2=s u b p l o t ( 2, 1, 2 ) ; M2 = l i n s p a c e ( LSmstar , LSmstar , 3 1 ) ; PLS = LSEnergy (X1, Y1,M2) ; p l o t ( f2,m2, PLS, + ) ; 61 x l a b e l ( m ) ; y l a b e l ( E n e r g y ) ; t i t l e ( L2 E n e r g y v e r s u s s l o p e ) ; Let s look at this sample data which is in the file FittingData.txt. Listing 2. Sample Data

4 MatLab Code for Simple Convex Minimization 4/ We load this data and plot it as follows: Listing 3. Plotting Raw Data >> Data = l o a d ( F i t t i n g D a t a. t x t ) ; 2 >> X = Data ( :, 1 ) ; >> Y = Data ( :, 2 ) ; >> p l o t (X,Y) ; >> Data = l o a d ( F i t t i n g D a t a. t x t ) ; >> X = Data ( :, 1 ) ; 7 >> Y = Data ( :, 2 ) ; >> p l o t (X,Y) ; >> x l a b e l ( T i m e ) ; >> y l a b e l ( T e m p e r a t u r e ) ; >> t i t l e ( R a w D a t a ) ; We see the data in Figure 1. Figure 1. Sample Data

5 MatLab Code for Simple Convex Minimization 5/8 Then we run the minimization code: note the optimal k value is different for the E One and E LS minimizations. Note this run uses the raw data files X and Y and so is doing a different minimization than the one we do in the cooling project. This code performs both the least squares and the summed absolute value error on any data X and Y we give it. Listing 4. The Raw Data Models >> [ mstar, E, Df, LSmstar, LSE ] = LOneEnergy (X,Y) ; >> mstar mstar = >> LSmstar LSmstar = The code generates a nice picture of this in Figure 2. We focus on a small piece of the these respective energy function as otherwise the vertical scale is too large to let us see the minimum behavior well. We see the data in Figure 1. Figure 2. E One and E LS Minimization Results The returned variable Df contains the subgradient sets for each of the knot points in E One. Listing 5. Subgradients at the Knots

6 MatLab Code for Simple Convex Minimization 6/8 >> Df Df = Note knot 14 is the subgradient which contains 0 and so this is the location of the absolute minimum of E One. Of course, these results are not really what we want as they solve a different minimization problem than the cooling one. So now we need to apply this ideas to the cooling project. We need to create the right Y here. The code is this: Listing 6. Creating the right Y Data = l o a d ( F i t t i n g D a t a. t x t ) ; % g r a b t h e t i m e s X = Data ( :, 1 ) ; % g r a b t h e t e m p e r a t u r e s 5 Temp = Data ( :, 2 ) ; % s e t t h e ambient t e m p e r a t u r e A = ; % compute t h e t r a n s f o r m e d v a r i a b l e U % b u t c a l l i t Y 10 Y = l o g ( (Temp A). / ( Temp( 1 ) A) ) ; Now find the two models: Listing 7. Finding the two k values [ mstar, E, Df, LSmstar, LSE ] = LOneEnergy (X,Y) ; % g r a b t h e k v a l u e f o r t h e summed a b s o l u t e e r r o r mstar mstar = % g r a b t h e k v a l u e f o r t h e l e a s t s q u a r e s e r r o r LSmstar

7 MatLab Code for Simple Convex Minimization 7/8 LSmstar = We note these are quite close in value. Now compare the plots of raw data and the two models. The code will generate a plot of these m values. Save it for printing and then delete it as we want to generate plots for the new models now. Listing 8. Building the Two Models and Comparing % b u i l d t h e summed a b s o l u t e e r r o r model ModelL1 = A + (Temp( 1 ) A) exp ( mstar X) ; 3 % b u i l d t h e l e a s t s q u a r e s model ModelL2 = A + (Temp( 1 ) A) exp ( LSmstar X) ; % compare t h e p l o t s p l o t (X, Temp, X, ModelL1, X, ModelL2 ) ; We show the results in Figure 3. So after you generate this plot, save it for printing later. Figure 3. The Raw Data and the Two Models 3. Extra Credit You can earn up to 25 points in extra credit for Project Two by doing the folloing: Exercise 3.1 Completely explain the MatLab code you have been provided with here. You need to figure out what I am doing to calculate the subgradients.

8 MatLab Code for Simple Convex Minimization 8/8 Exercise 3.2 Use this code on the data from Project Two. You should get the same answer as the one you got when you did it by hand calculation.