I am trying to solve the following optimization problem
\begin{align} \min_{X_{1},X_{2},y_{11},y_{12},y_{21},y_{22}} \; \; p_{1}X_{1}+p_{2}X_{2}& \\ \text{s.t}\; \; X_{1}^{\beta} &= y_{11} + y_{12} \hspace{2.5pt}\notag \\ X_{2}^{\beta} &= y_{21} + y_{22} \notag \\ Y_{1} &= \frac{y_{11}}{c_{11}} + \frac{y_{21}}{c_{21}} \notag \\ Y_{2} &= \frac{y_{12}}{c_{12}} + \frac{y_{22}}{c_{22}} \notag \\ X_{1},&X_{2},y_{11},y_{12},y_{21},y_{22} > 0 \notag \\ \end{align}
with parameters $\beta \in (0,1)$, $p_{1},p_{2}>0,$ $Y_{1},Y_{2}>0$ and $c_{11},c_{12},c_{21},c_{22}>1$.
I am having lots of trouble finding the solution to this minimization problem, which in principle, looks easy. The first order conditions are
\begin{align} &(1)\hspace{10pt}X_{1} : p_{1} - \lambda_{1}\beta X_{1}^{\beta-1} = 0 \\ &(2)\hspace{10pt}X_{2} : p_{2} - \lambda_{2}\beta X_{2}^{\beta-1} = 0 \\ &(3)\hspace{10pt}y_{11} : \lambda_{1} - \frac{\mu_{1}}{c_{11}} = 0 \\ &(4)\hspace{10pt}y_{12} : \lambda_{1} - \frac{\mu_{2}}{c_{12}} = 0 \\ &(5)\hspace{10pt}y_{21} : \lambda_{2} - \frac{\mu_{1}}{c_{21}} = 0 \\ &(6)\hspace{10pt}y_{22} : \lambda_{2} - \frac{\mu_{2}}{c_{22}} = 0 \\ \end{align}
where $\lambda,\mu$ are the Lagrange multipliers. Am I missing a constraint or something? Is there a way to go forward?
One natural option seems to use equations (3) and (5), and (4) and (6) to find the following conditions
\begin{align*} (7)\hspace{10pt}\frac{p_{1}c_{11}}{X_{1}^{\beta-1}} = \frac{p_{2}c_{21}}{X_{2}^{\beta-1}} \end{align*}
and
\begin{align*} (8)\hspace{10pt}\frac{p_{1}c_{12}}{X_{1}^{\beta-1}} = \frac{p_{2}c_{22}}{X_{2}^{\beta-1}} \end{align*}
However, how can (7) and (8) hold simultaneously?
The method of Lagrange multiplier may fail in some cases.
Consider the following optimization problem: \begin{align*} &\min_{X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22}} ~ p_1X_1 + p_2X_2 \qquad\qquad (P)\\ &\mathrm{subject ~ to}\quad X_1^\beta = y_{11} + y_{12},\\ &\qquad\qquad\quad\ X_2^\beta = y_{21} + y_{22}, \\ &\qquad\qquad\quad\ Y_1 = y_{11}/c_{11} + y_{21}/c_{21},\\ &\qquad\qquad\quad\ Y_2 = y_{12}/c_{12} + y_{22}/c_{22},\\ &\qquad\qquad\quad\ X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22} \ge 0. \end{align*} In some cases, at minimum, some of $X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22}$ are zero (see the example later), so the method of Lagrange multiplier fails. We need the KKT conditions instead. By the way, if your constraints are $X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22} > 0$, in such cases, no minimum occurs, which is similar to: $f(x) = x^2$ has no minimum on $(0, 1]$.
An example:
For $p_1 = 2, p_2 = 3, \beta = 1/2, Y_1 = 3, Y_2 = 4, c_{11} = 3, c_{12} = 6, c_{21} = 7, c_{22} = 4$,
the optimization problem is given by \begin{align*} &\min_{X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22}} ~ 2X_1 + 3X_2 \qquad\qquad (P1)\\ &\mathrm{subject ~ to}\quad \sqrt{X_1} = y_{11} + y_{12},\\ &\qquad\qquad\quad\ \sqrt{X_2} = y_{21} + y_{22}, \\ &\qquad\qquad\quad\ Y_1 = y_{11}/3 + y_{21}/7,\\ &\qquad\qquad\quad\ Y_2 = y_{12}/6 + y_{22}/4,\\ &\qquad\qquad\quad\ X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22} \ge 0. \end{align*} The minimum occurs at $(X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22}) = (4356/25, 4356/25, 9, 21/5, 0, 66/5)$.
The KKT conditions are given by \begin{align*} 2 - \lambda_1\frac{1}{2\sqrt{X_1}} - s_1 &= 0, \\ 3 - \lambda_2\frac{1}{2\sqrt{X_2}} - s_2 &= 0, \\ \lambda_1 - \frac13 \mu_1 - s_3 &= 0, \\ \lambda_1 - \frac16\mu_2 - s_4 &= 0, \\ \lambda_2 - \frac17\mu_1 - s_5 &= 0, \\ \lambda_2 - \frac14\mu_2 - s_6 &= 0,\\ s_1 X_1 &= 0, \\ s_2 X_2 &= 0, \\ s_3 y_{11} &= 0, \\ s_4 y_{12} &= 0, \\ s_5 y_{21} &= 0, \\ s_6 y_{22} &= 0, \\ s_1, s_2, s_3, s_4, s_5, s_6 &\ge 0. \end{align*} You may solve this system to obtain the solution.
By the way, the problem (P) is equivalent to the following convex problem: \begin{align*} &\min_{X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22}} ~ p_1X_1 + p_2X_2 \qquad\qquad (P2)\\ &\mathrm{subject ~ to}\quad X_1^\beta \ge y_{11} + y_{12},\\ &\qquad\qquad\quad\ X_2^\beta \ge y_{21} + y_{22}, \\ &\qquad\qquad\quad\ Y_1 = y_{11}/c_{11} + y_{21}/c_{21},\\ &\qquad\qquad\quad\ Y_2 = y_{12}/c_{12} + y_{22}/c_{22},\\ &\qquad\qquad\quad\ X_1, X_2, y_{11}, y_{12}, y_{21}, y_{22} \ge 0. \end{align*} Note: We replace the constraints $X_1^\beta = y_{11} + y_{12}, X_2^\beta = y_{21} + y_{22}$ with $X_1^\beta \ge y_{11} + y_{12}, X_2^\beta \ge y_{21} + y_{22}$.
We used Matlab+CVX to solve the convex problem (P2).