= Objective function
{title2=$f(x)$}
An <objective function> is a <real-valued function> assigning a score to each decision under consideration. An <optimization> problem seeks a <feasible solution> maximizing or minimizing this score. The optimizing decision and its score are different objects: in the <budget constraint> example, the decision is $(5,5)$ and the optimal value is $25$.
Choosing an <objective function> expresses what matters. With $x+y=10$ and $x,y\geq0$, maximizing $x+y$ makes every <feasible point> equally good; maximizing $x$ selects $(10,0)$; maximizing $xy$ rewards balance. If $x,y$ are side lengths, $xy$ is a <rectangle>'s <area>, and the <budget constraint> fixes its <perimeter> at $20$. The same mathematics then says that a <square> encloses the largest <area> among these <rectangles>.
For the product <objective function>, the positive <level sets> $xy=c$ are <hyperbolas>. Each collects decisions with identical scores. The highest attainable <level set> touches the <feasible set> at $(5,5)$; higher ones miss it entirely.
\Image[/optimization-budget-product.png]
{title=Product contours meet the feasible budget segment, with a unique maximum of 25 at (5,5)}
The product $xy$ is neither a <convex function> nor a <concave function> on the positive quadrant: along $(t,t)$ it is $t^2$, while along $(t,10-t)$ it is $10t-t^2$. Yet its restriction to this <feasible set> is a <strictly concave function>. Thus the geometry of the allowed decisions can make an otherwise nonconcave <maximization problem> tractable. Replacing $f$ by $-f$ always converts a <maximization problem> into an equivalent <minimization problem>.
// Target: mathematical-optimization.bigb
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