Convex Optimization

Why convex problems are the ones we can actually solve — recognizing convexity, the standard problem families, Lagrange duality and KKT, and the algorithms that find the optimum.

Convex Sets & Functions

• Convex Sets, Hulls & Cones — Sets you cannot escape by walking straight: segments, hulls, and cones.
• Operations That Preserve Convexity — Certify convexity by decomposition — sums, maxima, affine maps, and perspectives, never a Hessian.
• Separating & Supporting Hyperplanes — Two disjoint convex sets can always be cut apart by a single flat blade.
• Convexity in Several Variables — The chord definition survives the jump to vectors; the tests become the gradient inequality and a positive semidefinite Hessian.
• The Conjugate Function — f^(y) is the largest gap between a line of slope y and the graph of f — a construction that returns a convex function whatever you feed it.
• Quasiconvex Functions — Every sublevel set an interval — weak enough to include and x , strong enough for bisection.
• Dual Cones — Every cone casts a shadow: the directions that lean the same way as all of it.

Problem Classes & Duality

• Convex Optimization Problems — Convex objective, convex inequalities, affine equalities — and then every local optimum is global.
• Quadratic Programs & QCQP — Minimize a convex quadratic over a polyhedron — and unlike a linear program, the optimum need not be at a corner.
• Second-Order Cone & Semidefinite Programs — Swap the componentwise ordering of a linear program for an ice-cream cone, or for the positive semidefinite matrices.
• Geometric Programming — Products of arbitrary powers look hopeless; substitute x = e^y and the whole problem turns convex.
• The Lagrange Dual — Price the constraints instead of enforcing them, and a concave lower bound appears under any problem at all.
• Strong Duality & Slater's Condition — Weak duality is free; closing the gap costs one strictly feasible point.

Optimality & Algorithms

• The KKT Conditions — Lagrange multipliers meet complementary slackness: four conditions that certify an optimum.
• Sensitivity & Shadow Prices — The multiplier you computed is a price: what one more unit of budget is worth.
• Descent Methods & Line Search — Any direction that points downhill will do — the work is deciding how far to step.
• Newton's Method for Minimization — The root-finder aimed at f' — one step solves any quadratic, and no change of coordinates can slow it down.
• The Barrier Method — Trade every hard wall for a smooth one, then let the wall sharpen.
• Norm Approximation & Regularization — Least squares is one choice of norm among several, and the choice decides who wins: the cluster or the outlier.

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