Ad-hoc Optimization

Consider the following optimization problem:

\begin{array}{rl} & \displaystyle \max_{p_1,p_2} f(p_1,p_2) = {p_1x + p_2y}\\[0.2in] \text{subject to:} & p_1^2+p_2^2 \leq 1. \end{array}

Let’s rewrite the objective function such that (p_1^\star,p_2^\star) is a solution of the original problem if and only if it is also a solution for the modified objective function. Denote F the new objective function:

\begin{array}{rl} \displaystyle F(p_1,p_2) &= (p_1x + p_2y)^2 \\[0.15in] &= p_1^2x^2 + 2 \cdot p_1x \cdot p_2y + p_2^2y^2 \\[0.15in] &= (p_1^2 + p_2^2)(x^2 + y^2) - p_1^2y^2 - p_2^2x^2 + 2 \cdot p_1x \cdot p_2y + p_2^2y^2 \\[0.15in] &\leq (x^2 + y^2) - (p_1y - p_2x)^2. \end{array}

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