A convex optimization problem is one in which the objective and constraint functions are convex, which means they satisfy the inequality
\(f_i(\alpha x+\beta y) \leq \alpha f_i(x)+\beta f_i(y)\)
for all \(x, y \in \mathbf{R}^n\) and all \(\alpha, \beta \in \mathbf{R}\) with \(\alpha+\beta=1, \alpha \geq 0, \beta \geq 0\).
Since any linear program is therefore a convex optimization problem, we can consider convex optimization to be a generalization of linear programming.
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