The problem of finding the set of pareto optimals for constraints and qualitative preferences together is of great interest to many application areas. It can be viewed as a preference constrained optimization problem where the goal is to find one or more feasible solutions that are not dominated by other feasible outcomes. Our work aims to enhance the current literature of the problem by providing solving methods targeting the problem in a dynamic environments. We target the problem with an eye on adopting and benefiting from the current constraint satisfaction techniques.
This article presents an innovative research project (sponsored by the National Science Foundation, the American Iron and Steel Institute, and the American Institute of Steel Construction) where computationally elegant algorithms based on the integration of a novel connectionist computing model, mathematical optimization, and a massively parallel computer architecture are used to automate the complex process of engineering design.
Existing approaches to online convex optimization (OCO) make sequential one-slot-ahead decisions, which lead to (possibly adversarial) losses that drive subsequent decision iterates. Their performance is evaluated by the so-called regret that measures the difference of losses between the online solution and the best yet fixed overall solution in hindsight. The present paper deals with online convex optimization involving adversarial loss functions and adversarial constraints, where the constraints are revealed after making decisions, and can be tolerable to instantaneous violations but must be satisfied in the long term. Performance of an online algorithm in this setting is assessed by: i) the difference of its losses relative to the best dynamic solution with one-slot-ahead information of the loss function and the constraint (that is here termed dynamic regret); and, ii) the accumulated amount of constraint violations (that is here termed dynamic fit). In this context, a modified online saddle-point (MOSP) scheme is developed, and proved to simultaneously yield sub-linear dynamic regret and fit, provided that the accumulated variations of per-slot minimizers and constraints are sub-linearly growing with time. MOSP is also applied to the dynamic network resource allocation task, and it is compared with the well-known stochastic dual gradient method. Under various scenarios, numerical experiments demonstrate the performance gain of MOSP relative to the state-of-the-art.