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Which optimization technique/algorithm can be used to solve such problems? I want to know the name of a technique because some problems I need to solve are more complex than this one.

\begin{align} \max_{x_1,x_2}\quad \log_{2}(1+\dfrac{x_1}{x_2+0.1})+\log_{2}(1+\dfrac{x_2}{x_1+0.1})\\ s.t\quad\quad \log_{2}(1+\dfrac{x_2}{x_1+0.1})\geq0.1\\\quad \log_{2}(1+\dfrac{x_1}{x_2+0.1})\geq0.1 \\x_1\geq0,x_2\geq0\\ x_1\leq5,x_2\leq5 \end{align}

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    please don't tell me that the optimal solution is given by this \begin{align} x_{2}=(2^{0.1}-1)5.1\\ x_{1}=5 \end{align} Thell me the name of an optimization technique that can solve this. – 2017-02-11
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    I want an answer like, "use sequential quadratic programming","use particle swarm optimization" etc any technique that is sophisticated like SQP. – 2017-02-11
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    I don't want the a answer of this problem. I want the name of an optimization technique that would work on this problem and problems like this. – 2017-02-11

1 Answers 1

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Take the following inequality constraint

$$\log_{2} \left( 1 + \dfrac{x_j}{x_i+0.1} \right) \geq 0.1$$

and rewrite it as follows

$$\dfrac{x_j}{x_i+0.1} \geq 2^{0.1} - 1$$

Given the nonnegativity constraint $x_i \geq 0$, we can multiply both sides by $x_i+0.1$, which yields the following linear inequality

$$\left(2^{0.1} - 1\right) x_i - x_j \leq -\left(2^{0.1} - 1\right) 0.1$$

Thus, we conclude that the feasible region is a polygon, namely, a quadrilateral.

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  • Parametrizing each of the $4$ line segments that form the quadrilateral, we can find the maximum of the objective function on the boundary of the feasible region.

  • Computing the gradient of the objective function and finding whether it vanishes in the interior of the feasible region and, if so, where it does vanish, we can find the maximum of the objective function in the interior of the feasible region (when it actually does exist).

Taking the maximum of these two maxima, we obtain the desired maximum.