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Consider the (non-linear) optimization problem ($P$)

$$max \quad3x_1 + 4x_2$$

$$s.t. \quad x_1^2 + x_2^2 \leq 25$$

$$ \quad x_1,x_2 \geq 0$$

Formulate the Lagrangian function $\varTheta(y)$ and solve it for fixed $y>0$

This is the function that I obtained. $\varTheta(y)=25y_1 + max\{(3-y_1x_1)x_1 + (4-y_1x_2)x_2; \quad x_1,x_2\geq 0\}$

However, Since the coefficents in the $max$ function still have $x_i$'s in them, I am not sure how to proceed. Could anyone please help me out?

Community
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dreamer
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1 Answers1

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Since the objective function is increasing in the domain you just have to check the boundary an the problem reduces to $$max_{x_1,x_2}\quad 3x_1+4x_2$$ $$s.t.\quad x_1^2+x_2^2=25$$ which has the Lagrangian $$\varTheta(x_1,x_2,\lambda)=3x_1+4x_2-\lambda(x_1^2+x_2^2-25)$$ By applying first order conditions $$\frac{\partial \varTheta}{\partial x_1}=3-2\lambda x_1=0$$ $$\frac{\partial \varTheta}{\partial x_2}=4-2\lambda x_2=0$$ $$\frac{\partial \varTheta}{\partial \lambda}=-x_1^2-x_2^2+25=0$$ which can be solved by $x_1=3$, $x_2=4$ and $\lambda=1/2$.

AnilB
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  • Ah, I see. I thought the problem had to be solved by first constructing a dual problem but that seems impossible. How stupid that I did not think of this. Thank you for your help :))! – dreamer May 16 '13 at 20:31
  • You are welcome – AnilB May 16 '13 at 20:34