User Name Remember Me? Password

 Calculus Calculus Math Forum

 February 15th, 2015, 03:28 PM #1 Newbie   Joined: Feb 2015 From: London Posts: 1 Thanks: 0 Inequality constraint NLP problem Hello, Regarding subject, I have a difficulty with following NLP problem. Max f(x) = X1 / (X2+1) s.t. X1 - X2 <= 2 X1>=0 X2>=0 Actually, I could get the answer with simple graphical way that optima is 2 at X = [2;0] However, I could not get the solution with Lagrange multiplier method. After partial differentiation of L(X1, X2, lamda), I could get 3 equations and 3 variables, but two equations are parallel so it does not give me a solution. Could anyone solve this problem? And also, could I get Kuhn-Tucker conditions (KKT) of this solution? In addition, can't we say this problem is convex? Thank you. Last edited by skipjack; February 15th, 2015 at 11:26 PM. February 15th, 2015, 06:27 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra To maximise a quotient, you minimise the denominator and maximise the numerator. Thus we want $x_2 = 0$ and then $x_1 - x_2 \le 2 \implies x_1 = 2$. Of course, you might suggest that we could add $a \gt 0$ to each of these values and still satisfy the inequalities. This would work if$${x_1 + a \over x_2 + 1 + a} \gt {x_1 \over x_2 + 1}$$ Since we have that $x_2 = x_1 - 2$ we then have \begin{aligned}{x_1 + a \over x_1 - 1 + a} &\gt {x_1 \over x_1 - 1} \\ (x_1 - 1)(x_1 + a) &\gt x_1(x_1 - 1 + a) \\ x_1^2 + (a-1)x_1 -a &\gt x_1^2 + (a - 1)x_1 \\ a \lt 0\end{aligned} But we have already stated the $a \gt 0$, and indeed $a \lt 0$ violates the minimum value of $x_2$. So $(x_1, x_2) = (2,0)$ is the only solution. February 15th, 2015, 11:51 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,942 Thanks: 2210 As 0 $\leqslant$ X1 $\leqslant$ 2 + X2, one must take X1 = 2 + X2 to maximize X1/(X2 + 1). That gives f(x) = (2 + X2)/(X2 + 1) = 2 - X2/(X2 + 1). As X2 $\geqslant$ 0, the maximum value of f(x) is 2, achieved when X2 = 0 (and X1 = 2). It would have been better to write f(X1, X2) instead of f(x). Tags constraint, inequality, nlp, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post helloprajna Economics 0 February 17th, 2013 11:04 PM palarce Economics 0 May 7th, 2012 05:59 AM Gekko Calculus 3 June 12th, 2010 09:16 AM Justin Lo Applied Math 0 December 8th, 2009 02:40 AM shack Linear Algebra 3 December 17th, 2007 01:11 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.      