Robust Quadratic Programming Research Articles

Robust Quadratic Programming for MDPs with uncertain observation noise

The problem of Markov decision processes (MDPs) with uncertain observation noise has rarely been studied. This paper proposes a Robust Quadratic Programming (RQP) approach to approximate Bellman equation solution. Besides efficiency, the proposed algorithm exhibits great robustness against uncertain observation noise, which is essential in real world applications. We further represent the solution into kernel forms, which implicitly expands the state-encoded feature space to higher or even infinite dimensions. Experimental results well justify its efficiency and robustness. The comparison with different kernels demonstrates its flexibility of kernel selection for different application scenarios.

A Numerically Robust Mixed-Integer Quadratic Programming Solver for Embedded Hybrid Model Predictive Control

Abstract The deployment of hybrid model predictive control (MPC) in practical applications requires primarily an efficient and robust on-line Mixed-Integer Quadratic Programming (MIQP) solver that runs in real time. In this paper we propose a new algorithm for solving MIQP problems which is particularly tailored to solve small-scale MIQPs, such as those that arise in embedded hybrid MPC applications. The algorithm couples a branch and bound (B&B) scheme with a recently proposed numerically robust Quadratic Programming (QP) solver based on nonnegative least squares (NNLS) and proximal-point iterations. The resulting MIQP solver supports positive semidefinite Hessian matrices, often appearing in hybrid MPC formulations, and warm starts with respect to both binary and real variables. We show that the speed of execution of our solver is comparable with state-of-the-art commercial solvers, while it is relatively simple to code in an embedded control system.

Immunizing Conic Quadratic Optimization Problems Against Implementation Errors

We show that the robust counterpart of a convex quadratic constraint with ellipsoidal implementation error is equivalent to a system of conic quadratic constraints. To prove this result we first derive a sharper result for the S-lemma in case the two matrices involved can be simultaneously diagonalized. This extension of the S-lemma may also be useful for other purposes. We extend the result to the case in which the uncertainty region is the intersection of two convex quadratic inequalities. The robust counterpart for this case is also equivalent to a system of conic quadratic constraints. Results for convex conic quadratic constraints with implementation error are also given. We conclude with showing how the theory developed can be applied in robust linear optimization with jointly uncertain parameters and implementation errors, in sequential robust quadratic programming, in Taguchi’s robust approach, and in the adjustable robust counterpart.

Interval-parameter robust quadratic programming for water quality management under uncertainty

Effective planning of water quality management is important for facilitating sustainable socio-economic development in watershed systems. An interval-parameter robust quadratic programming (IRQP) method is developed by incorporating techniques of robust programming and interval quadratic programming within a general optimization framework. The IRQP improves upon existing quadratic programming methods, and can tackle uncertainties presented as interval numbers and fuzzy sets as well as their combinations. Moreover, it can deal with nonlinearities in the objective function such that economies-of-scale effects can be reflected. The developed method is applied to a case study of a water quality management under uncertainty. A number of decision alternatives are generated based on the interval solutions as well as the projected applicable conditions. They represent multiple decision options with various environmental and economic considerations. Willingness to accept a low economic revenue will guarantee satisfying the water quality requirements. A strong desire to acquire a high benefit will run the risk of violating environmental criteria.