Optimization

Necessary optimality conditions for constrained optimization via Lagrangian gradient and complementary slackness

A function is convex when the chord between any two points lies above the graph

For a convex function, the image of a weighted sum is at most the weighted sum of images

At a constrained extremum, the gradient of the objective is proportional to the gradient of the constraint

Every linear program has a dual whose optimal value equals the primal's — strong duality

For a convex-concave function on compact sets, the min-max and max-min values coincide

Every finite strategic-form game has at least one mixed Nash equilibrium

For L-smooth convex functions, gradient descent with step 1/L converges geometrically to the minimum