The two biggest challenges in solving practical optimization problems are computational intractability, and the presence of uncertainty: most problems are either NP-hard, or have incomplete input data which makes an exact computation impossible. Recently, there has been a huge progress in our understanding of intractability, based on spectacular algorithmic and lower bound techniques. For several problems, especially those with only local constraints, we can design optimum approximation algorithms that are provably the best possible. However, typical optimization problems usually involve complex global constraints and are much less understood. The situation is even worse for coping with uncertainty. Most of the algorithms are based on ad-hoc techniques and there is no deeper understanding of what makes various problems easy or hard. This proposal describes several new directions, together with concrete intermediate goals, that will break important new ground in the theory of approximation and online algorithms. The particular directions we consider are (i) extend the primal dual method to systematically design online algorithms, (ii) build a structural theory of online problems based on work functions, (iii) develop new tools to use the power of strong convex relaxations and (iv) design new algorithmic approaches based on non-constructive proof techniques. The proposed research is at the cutting edge of algorithm design, and builds upon the recent success of the PI in resolving several longstanding questions in these areas. Any progress is likely to be a significant contribution to theoretical computer science and combinatorial optimization.