Scheduling under Linear Constraints

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.Comment: 21 page

Trajectory Optimization and Guidance Design by Convex Programming

The field of aerospace guidance and control has recently been evolving from focusing on traditional laws and controllers to numerical algorithms with the aim of achieving onboard applications for autonomous vehicle systems. However, it is very difficult to perform complex guidance and control missions with highly nonlinear dynamic systems and many constraints onboard. In recent years, an emerging trend has occurred in the field of Computational Guidance and Control (CG&C). By taking advantage of convex optimization and highly efficient interior point methods, CG&C allows complicated guidance and control problems to be solved in real time and offers great potential for onboard applications. With the significant increase in computational efficiency, convex-optimization-based CG&C is expected to become a fundamental technology for system autonomy and autonomous operations. In this dissertation, successive convex approaches are proposed to solve optimal control programs associated with aerospace guidance and control, and the emphasis is placed on potential onboard applications. First, both fuel-optimal and time-optimal low-thrust orbit transfer problems are investigated by a successive second-order cone programming method. Then, this convex method is extended and improved to solve hypersonic entry trajectory optimization problems by taking advantage of line-search and trust-region techniques. Finally, the successive convex approach is modified to the design of autonomous entry guidance algorithms. Simulation results indicate that the proposed methodologies are capable of generating accurate solutions for low-thrust orbit transfer problems and hypersonic entry problems with fast computational speed. The proposed methods have great potential for onboard applications

The cohomology rings of Hilbert schemes via Jack polynomials

In this note, generalizing earlier work of Nakajima and Vasserot, we study the (equivariant) cohomology rings of Hilbert schemes of certain toric surfaces and establish their connections to Fock space and Jack polynomials.Comment: Appeared in a CRM proceedings edited by Hurtubise and Markma