https://www.chenk.top/en/tags/portfolio-optimization/Recent content in Portfolio Optimization on Chen Kai BlogHugoenFri, 30 Sep 2022 09:00:00 +0000https://www.chenk.top/en/optimization-theory/12-discrete-global-optimization/Fri, 30 Sep 2022 09:00:00 +0000https://www.chenk.top/en/optimization-theory/12-discrete-global-optimization/<p>The first eleven articles in this series tackled <strong>continuous convex</strong> problems (or convex relaxations of non-convex ones). This final article addresses two harder regimes:</p> <ul> <li><strong>Discrete optimization</strong>: variables take integer or combinatorial values. The feasible set is a finite (but exponentially large) collection of points. Linear and convex tools no longer apply directly because there are no derivatives across the integer lattice.</li> <li><strong>Global non-convex optimization</strong>: variables are continuous but the function has many local minima, and we want the <em>global</em> one. Methods like Newton and L-BFGS only find local minima.</li> </ul> <p>Both regimes share a key feature: <strong>provably optimal algorithms are exponential</strong> in the worst case. Practical solutions come from (a) exact algorithms with smart pruning (branch-and-bound) and (b) heuristics that find good (but not optimal) solutions in polynomial time.</p>