On Repetitive Scenario Design

Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses $N$ scenarios (design samples), followed by randomized feasibility phase that uses $N_o$ test samples on the scenario solution. We give a full and exact probabilistic characterization of the number of iterations required by the RSD approach for returning a solution, as a function of $N$, $N_o$, and of the desired levels of probabilistic robustness in the solution. This novel approach broadens the applicability of the scenario technology, since the user is now presented with a clear tradeoff between the number $N$ of design samples and the ensuing expected number of repetitions required by the RSD algorithm. The plain (one-shot) scenario design becomes just one of the possibilities, sitting at one extreme of the tradeoff curve, in which one insists in finding a solution in a single repetition: this comes at the cost of possibly high $N$. Other possibilities along the tradeoff curve use lower $N$ values, but possibly require more than one repetition

Robust Model Predictive Control via Scenario Optimization

This paper discusses a novel probabilistic approach for the design of robust model predictive control (MPC) laws for discrete-time linear systems affected by parametric uncertainty and additive disturbances. The proposed technique is based on the iterated solution, at each step, of a finite-horizon optimal control problem (FHOCP) that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances, followed by a specific command selection rule implemented in a receding horizon fashion. The scenario FHOCP is always convex, also when the uncertain parameters and disturbance belong to non-convex sets, and irrespective of how the model uncertainty influences the system's matrices. Moreover, the computational complexity of the proposed approach does not depend on the uncertainty/disturbance dimensions, and scales quadratically with the control horizon. The main result in this paper is related to the analysis of the closed loop system under receding-horizon implementation of the scenario FHOCP, and essentially states that the devised control law guarantees constraint satisfaction at each step with some a-priori assigned probability p, while the system's state reaches the target set either asymptotically, or in finite time with probability at least p. The proposed method may be a valid alternative when other existing techniques, either deterministic or stochastic, are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance-constrained optimization problem.Comment: This manuscript is a preprint of a paper accepted for publication in the IEEE Transactions on Automatic Control, with DOI: 10.1109/TAC.2012.2203054, and is subject to IEEE copyright. The copy of record will be available at http://ieeexplore.ieee.or

Stochastic model predictive control of LPV systems via scenario optimization

A stochastic receding-horizon control approach for constrained Linear Parameter Varying discrete-time systems is proposed in this paper. It is assumed that the time-varying parameters have stochastic nature and that the system's matrices are bounded but otherwise arbitrary nonlinear functions of these parameters. No specific assumption on the statistics of the parameters is required. By using a randomization approach, a scenario-based finite-horizon optimal control problem is formulated, where only a finite number M of sampled predicted parameter trajectories (‘scenarios') are considered. This problem is convex and its solution is a priori guaranteed to be probabilistically robust, up to a user-defined probability level p. The p level is linked to M by an analytic relationship, which establishes a tradeoff between computational complexity and robustness of the solution. Then, a receding horizon strategy is presented, involving the iterated solution of a scenario-based finite-horizon control problem at each time step. Our key result is to show that the state trajectories of the controlled system reach a terminal positively invariant set in finite time, either deterministically, or with probability no smaller than p. The features of the approach are illustrated by a numerical example

Nanoimprinting of Photonic Devices for Visible Light Applications

The science of photonics is increasingly enabling the discovery of unprecedented proprieties that stem from the interaction of light with nanostructured matter. On the other hand, nanotechnology provides a key support to the research in photonics. In particular nanoimprint lithography (NIL) proved important to accelerate the development and prototyping of novel photonic device concepts. In this thesis, NIL was analyzed as a suitable, flexible, low-cost lithography solution for photonics. The purpose of this work is to research and evaluate novel nanofabrication technologies, materials and device concepts to support and innovate the fields of photonics and nanotechnology. In doing so, different implementations of NIL were experimentally investigated on the imprint of novel photonic devices into purely organic and hybrid organic-inorganic sol-gel materials. Two main research themes on printable photonics are explored in parallel. The first concerns the development and testing of nanoimprint technologies to pursue the fabrication of progressively more challenging device concepts, from 2D to 3D. As a second research theme, the idea to couple top-down nano-patterning approaches (NIL) with bottom-up functionalities that emerge from engineering material proprieties at the chemistry level is put forward. To explore these research courses, two photonic devices were designed, fabricated and tested: an integrated holographic planar circuit for on-chip spectroscopy, and a photonic crystal slab printed out of a functional, high-refractive index material. The amount of integration, complexity and variety of the printed optical components presented here allow us to extend the validity of the work to an even broader range of photonic devices. This work advances the field of printable photonics and demonstrates its leverage to innovation, which encompasses several scientific fields

Parallel block coordinate minimization with application to group regularized regression

This paper proposes a method for parallel block coordinate-wise minimization of convex functions. Each iteration involves a first phase where n independent minimizations are performed over the n variable blocks, followed by a phase where the results of the first phase are coordinated to obtain the whole variable update. Convergence of the method to the global optimum is proved for functions composed of a smooth part plus a possibly non-smooth but separable term. The method is also proved to have a linear rate of convergence, for functions that are smooth and strongly convex. The proposed algorithm can give computational advantage over the more standard serial block coordinate-wise minimization methods, when run over a parallel, multi-worker, computing architecture. The method is suitable for regularized regression problems, such as the group Lasso, group Ridge regression, and group Elastic Net. Numerical tests are run on such types of regression problems to exemplify the performance of the proposed method

Optimal Dynamic Asset Allocation with Lower Partial Moments Criteria and Affine Policies

This paper discusses an optimization-based approach for solving multi-period dynamic asset allocation problems using empirical asymmetric measures of risk. Three features distinguish the proposed approach from the mainstream ones. First, our approach is non parametric, in the sense that it does not require explicit estimation of the parameters of a statistical model for the returns distribution: the approach relies directly on data (the scenarios) generated by an oracle which may include expert knowledge along with a standard stochastic return model. Second, it employs affine decision policies, which make the multi-period formulation of the problem amenable to an efficient convex optimization format. Third, it uses asymmetric, unilateral measures of risk which, unlike standard symmetric measures such as variance, capture the fact that investors are usually not averse to return deviations from the expected target, if these deviations actually exceed the target

A guaranteed-convergence framework for passivity enforcement of linear macromodels

Passivity enforcement is a key step in the extraction of linear macromodels of electrical interconnects and packages for Signal and Power Integrity applications. Most state-of-the-art techniques for passivity enforcement are based on suboptimal or approximate formulations that do not guarantee convergence. We introduce in this paper a new rigorous framework that casts passivity enforcement as a convex non-smooth optimization problem. Thanks to convexity, we are able to prove convergence to the optimal solution within a finite number of steps. The effectiveness of this approach is demonstrated through various numerical example

Repetitive Scenario Design

Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses N scenarios (design samples), followed by randomized feasibility phase that uses No test samples on the scenario solution. We give a full and exact probabilistic characterization of the number of iterations required by the RSD approach for returning a solution, as a function of N, No, and of the desired levels of probabilistic robustness in the solution. This novel approach broadens the applicability of the scenario technology, since the user is now presented with a clear tradeoff between the number N of design samples and the ensuing expected number of repetitions required by the RSD algorithm. The plain (one-shot) scenario design becomes just one of the possibilities, sitting at one extreme of the tradeoff curve, in which one insists in finding a solution in a single repetition: this comes at the cost of possibly high N. Other possibilities along the tradeoff curve use lower N values, but possibly require more than one repetition