164(Summer10)_MidtermSolutions

# 164(Summer10)_MidtermSolutions - MATH 164 Lecture 1 Summer...

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MATH 164 - Lecture 1 - Summer 2010 Midterm Solutions - July 13, 2010 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 5 problems for a total of 100 points. POINTS: 1. 2. 3. 4. 5. TOTAL: 1

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2 1. (20 points) Let x / be a local minimizer for a convex programming problem. Show that x / is also a global minimizer. Solution. Use contraction, let x / be a local minimizer, but not a global minimizer. Suppose there exists some point y 2 S , that f ( y ) < f ( x / ). Let 0 < ﬁ < 1, then f ( ﬁx / +(1 ¡ ) y ) ﬁf ( x / )+(1 ¡ ) f ( y ) < ﬁf ( x / )+(1 ¡ ) f ( x / ) = f ( x / ) : Which means if ! 1, there are points z = ﬁx / + (1 ¡ ) y , z 2 S , and arbitrarily close to x / , f ( z ) < f ( x / ). This contradicts the deﬂnition of the local minimizer. Thus the assumption isn’t correct, x / must be a global minimizer. /
3 2. (25 points) Suppose „ x is a feasible point for the program minimize f ( x ) = x 2 1 + x 2 + x 3 subject to x 1 ¡ 2 x 2 + 3 x 3 = 6 2 x 1 + x 2 ¡ x 3 3 x 2 ¡ 2 x 3 • ¡ 2 x 2 ;x 3 0 (a) Formulate the criterium for p to be a feasible direction at „ x (b) Find all feasible directions for „ x = (3 ; 0 ; 1) T (c) Determine whether p = (1 ; ¡ 7 ; ¡ 5) T is a feasible direction at „ x = (0 ; 15 ; 12). If yes, ﬂnd the maximal step

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164(Summer10)_MidtermSolutions - MATH 164 Lecture 1 Summer...

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