Weak Sharp Minima for Robust Optimization Problems

The weak sharp minima for optimization problems plays an important role in the stability analysis and sensitivity of solution set-valued mapping (that is, analyze the behavior of solution set-mapping when the data varies) and finite convergence analysis of numerical algorithms (that is, the sequence generated by the numerical algorithm converges after a finite number of iterates). It has been extensively studied for several optimization and its related problems, where perfect information is often assumed (that is, accurate values for the input quantities or parameters), despite the reality that such precise knowledge is rarely available in many real-world optimization problems, such as, petroleum engineering design and data analysis of natural gas engineering. Robust optimization provides a powerful method to deal with the issues of data uncertainty. In this project, we shall mainly use the techniques from robust optimization to study the weak sharp minima for conic optimization problems, where the uncertain parameter is assumed to be in a prescribed uncertainty set. For this, we intend to introduce the concepts of local / bounded / global weak sharp minima for robust conic optimization problems. Then, by using the subdifferential, directional derivative and normal cone in sense of convex analysis, we shall give some complete characterizations of these concepts for robust convex conic optimization problems. By using the tools from convex analysis and variational analysis, we intend to study the local / bounded / global weak sharp minima for robust DC (difference of two convex functions) conic optimization problems with convex constraints. We intend to use our results on weak sharp minima to design the algorithms for the robust counterparts of uncertain optimization problems and to analyze finite convergence of these algorithms. We shall try to provide optimal designs to petroleum engineering or / and natural gas engineering based on the obtained algorithms.