mixed integer programming

I. Prodan, Stoican, F., and Grotli, E., “Some remarks on potential field constructions in a multi-obstacle environment”, in 10th IFAC Conference on Control Applications in Marine Systems, Trondheim, Norway, 2016.Abstract
This paper addresses a novel combination between mixed-integer representations and potential field constructions for typical multi-agent marine control problems. First, we prove that for any kind of repulsive functions applied over a function which we denote as sum function, the feasible domain is piece-wise affine (PWA). Next, concepts like hyperplane arrangements together with potential field approaches are used for providing an efficient description of the feasible non-convex domain. This combination offers an original and beneficent computation of control laws under non-convex constraints. Simulation results over a common application of obstacle avoidance, which can be extended for unmanned surface vehicles, prove the effectiveness of the proposed approach.
Fault tolerant control based on set-theoretic methods
F. Stoican, “Fault tolerant control based on set-theoretic methods”, École Superieure d'Électricite (SUPELEC) , 2011. Online archiveAbstract
The scope of the thesis is the analysis and design of fault tolerant control (FTC) schemes through the use of set-theoretic methods. In the framework of multisensor schemes, the faults appearance and the modalities to accurately detect them are investigated as well as the design of control laws which assure the closed-loop stability. By using invariant/contractive sets to describe the residual signals, a fault detection and isolation (FDI) mechanism with reduced computational demands is implemented based on set-separation. A dual mechanism, implemented by a recovery block, which certificates previously fault-affected sensors is also studied. From a broader theoretical perspective, we point to the conditions which allow the inclusion of FDI objectives in the control law design. This leads to static feedback gains synthesis by means of numerically attractive optimization problems. Depending on the parameters selected for tuning, is shown that the FTC design can be completed by a reference governor or a predictive control scheme which adapts the state trajectory and the feedback control action in order to assure FDI. When necessary, the specific issues originated by the use of set-theoretic methods are detailed and various improvements are proposed towards: invariant set construction, mixed integer programming (MIP), stability for switched systems (dwell-time notions).
M. - I. Strutu, Stoican, F., Prodan, I., Popescu, D., and Olaru, S., “A characterization of the relative positioning of mobile agents for full sensorial coverage in an augmented space with obstacles”, in Proceedings of the 21st Mediterranean Conference on Control and Automation, Platania-Chania, Crete, Grecee, 2013, p. 936-941.Abstract
This paper addresses the coverage problem for a collection of agents and fixed obstacles (e.g., the “gallery” and the “patrolling” problems). A collection of sufficient conditions over the positions of the agents are provided such that whenever these are verified there is no “blind” region in the feasible space. These conditions are expressed by making use of hyperplane arrangements which lead to a mixed-integer formulation. Practical applications regarding the coverage problem inside an augmented space with obstacles validate these concepts and provide an efficient implementation (in terms of computing power).
F. Stoican, Prodan, I., and Olaru, S., “Hyperplane Arrangements in Mixed-Integer Programming Techniques. Collision avoidance application with Zonotopic Sets”, in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, 2013, p. 3155-3160.Abstract
The current paper addresses the problem of minimizing the computational complexity of optimization problems with non-convex and possibly non-connected feasible region of polyhedral type. Using hyperplane arrangements and Mixed-Integer Programming we provide an efficient description of the feasible region in the solution space. Moreover, we exploit the geometric properties of the hyperplane arrangements and adapt this description in order to provide an efficient solution of the mixed-integer optimization problem. Furthermore, a zonotopic representation of the sets appearing in the problem is considered. The advantages of this representation are highlighted and exploited through proof of concepts illustrations as well as simulation results.
F. Stoican, Prodan, I., and Olaru, S., “On the hyperplanes arrangements in mixed-integer techniques”, in Proceedings of the 30th American Control Conference, San Francisco, California, USA, 2011, p. 1898–1903.Abstract
This paper is concerned with the improved constraints handling in mixed-integer optimization problems. The novel element is the reduction of the number of binary variables used for expressing the complement of a convex (polytopic) region. As a generalization, the problem of representing the complement of a possibly non-connected union of such convex sets is detailed. In order to illustrate the benefits of the proposed improvements, a practical implementation, the problem of obstacle avoidance using receding horizon optimization techniques is considered.
F. Stoican, Prodan, I., and Olaru, S., “Enhancements on the hyperplane arrangements in mixed integer techniques”, in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, USA, 2011, p. 3986–3991.Abstract
The current paper addresses the problem of optimizing a cost function over a non-convex and possibly non-connected feasible region. A classical approach for solving this type of optimization problem is based on Mixed integer technique. The exponential complexity as a function of the number of binary variables used in the problem formulation highlights the importance of reducing them. Previous work which minimize the number of binary variables is revisited and enhanced. Practical limitations of the procedure are discussed and a typical control application, the control of Multi-Agent Systems is exemplified.
M. Hovd and Stoican, F., “On the design of exact penalty functions for MPC using mixed integer programming”, International Journal of Computers & Chemical Engineering, vol. 70, p. 104-113, 2014.Abstract
Soft constraints and penalty functions are commonly used in MPC to ensure that the optimization problem has a feasible solution, and thereby avoid MPC controller failure. On the other hand, soft constraints may allow for unnecessary violations of the original constraints, i.e., the constraints may be violated even if a valid solution that does not violate any constraints exists. The paper develops procedures for the minimizing (according to some norm) of the Lagrange multipliers associated with a given mp-QP problem, assumed to originate from an MPC problem formulation. To this end the LICQ condition is exploited in order to efficiently formulate the optimization problem, and thereby improve upon existing mixed integer formulations and enhance the tractability of the problem. The results are used to design penalty functions such that corresponding soft constraints are made exact, that is, the original (hard) constraints are violated only if there exists no solution where all constraints are satisfied.
I. Prodan, Stoican, F., Olaru, S., and Niculescu, S. I., “Enhancements on the hyperplanes arrangements in mixed integer techniques”, Journal of Optimization Theory and Applications, vol. 154, no. 2, p. 549-572, 2012.Abstract
This paper is concerned with improvements in constraints handling for mixed-integer optimization problems. The novel element is the reduction of the number of binary variables used for expressing the complement of a convex (polytopic) region. As a generalization, the problem of representing the complement of a possibly not connected union of such convex sets is detailed. In order to illustrate the benefits of the proposed improvements, a typical control application, the control of multiagent systems using receding horizon optimization techniques, is considered.
Mixed-Integer Programming Techniques in Distributed MPC Problems
I. Prodan, Stoican, F., Olaru, S., Stoica, C. N., and Niculescu, S. I., “Mixed-Integer Programming Techniques in Distributed MPC Problems”, in Distributed Model Predictive Control Made Easy, J. M. Maestre and Negenborn, R. R. New York: Springer, 2013, p. 273-288. Online infoAbstract
This chapter proposes a distributed approach for the resolution of a multiagent problem under collision and obstacle avoidance conditions. Using hyperplane arrangements and mixed integer programming, we provide an efficient description of the feasible region verifying the avoidance constraints. We exploit geometric properties of hyperplane arrangements and adapt this description to the distributed scheme in order to provide an efficient Model Predictive Control (MPC) solution. Furthermore,we prove constraint validation for a hierarchical ordering of the agents.