A subgradient is a generalization of the derivative for non-differentiable convex functions, providing a way to characterize their slopes at points in their domain. This concept allows for the optimization of functions that are not smooth, as it captures the idea of direction for improvement in terms of function values. Subgradients are particularly useful in establishing optimality conditions for convex optimization problems, as they offer necessary and sufficient conditions for finding optimal solutions in scenarios where traditional gradients may not exist.
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