Convex optimization formulation

  • How do you optimize a convex function?

    Unconstrained convex optimization can be easily solved with gradient descent (a special case of steepest descent) or Newton's method, combined with line search for an appropriate step size; these can be mathematically proven to converge quickly, especially the latter method..

  • Compositions.
    A basic rule of convex analysis is that convexity is closed under composition with an affine mapping.
    This is part of the DCP ruleset as well: A convex, concave, or affine function may accept an affine expression (of compatible size) as an argument.

Quadratic fractional programming problem

Bilevel optimization is a special kind of optimization where one problem is embedded (nested) within another.
The outer optimization task is commonly referred to as the upper-level optimization task, and the inner optimization task is commonly referred to as the lower-level optimization task.
These problems involve two kinds of variables, referred to as the upper-level variables and the lower-level variables.

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