The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Real and Complex Analysis

This paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms. This includes the wide class of NP-hard control problems representable by means of parameter-dependent linear matrix inequalities (LMIs). It is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem (the scenario problem) whose solution is approximately feasible for the original (usually infinite) set of constraints, i.e., the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased. We provide an explicit and efficient bound on the number of samples required to attain a-priori specified levels of probabilistic guarantee of robustness. A rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial-time solution, if robustness is intended in the proposed risk-adjusted sense.

https://www.sheffield.ac.uk/polopoly_fs/1.60408!/file/campi_scenario-optimization.pdf

The scenario approach to robust control design

(1990). ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY. Electric Machines & Power Systems: Vol. 18, No. 2, pp. 209-210.

Energy function analysis for power system stability

Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance’ parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations. In this paper, we consider an alternative ‘randomized’ or ‘scenario’ approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

/pdf/uncertain-convex-programs-randomized-solutions-and-3vs4hcxix0.pdf

Uncertain convex programs: randomized solutions and confidence levels

In this paper we consider design of stabilizing controllers for linear systems subjected to persistent exogenous disturbances. The performance of the closed-loop system is characterized by the size of invariant (bounding) ellipsoid for the system state (output). The optimal (smallest) ellipsoid is shown to be "fragile" in the sense that small variations of its coefficients may lead to considerable degradation of performance or even to the loss of stability of the closed-loop system. We propose a method for constructing a "nonfragile" controller that tolerates variations of the parameters and remains optimal in the sense of the performance index adopted.

Invariance and nonfragility in the rejection of exogenous disturbances

A probabilistic point of view on peak effects in linear difference equations

The paper presents a quantitative explanation of failure of generic Monte Carlo techniques as applied to optimization problems of high dimensions. Deterministic grids are also discussed.

Why does Monte Carlo fail to work properly in high-dimensional optimization problems?

This is an attempt to discuss the following question: When is a random choice better than a deterministic one? That is, if we have an original deterministic setup, is it wise to exploit randomization methods for its solution? There exist numerous situations where the positive answer is obvious; e.g., stochastic strategies in games, randomization in experiment design, randomization of inputs in identification. Another type of problems where such approach works successfully relates to treating uncertainty, see Tempo R., Calafiore G., Dabbene F., “Randomized algorithms for analysis and control of uncertain systems,” Springer, New York, 2013. We will try to focus on several research directions including optimization problems with no uncertainty and compare known deterministic methods with their stochastic counterparts such as random descent, various versions of Monte Carlo etc., for convex and global optimization. We survey some recent results in the field and ascertain that the situation can be very different.

Randomization in Robustness, Estimation, and Optimization

A theorem is formulated that gives an exact probability distribution for a linear function of a random vector uniformly distributed over a ball in n-dimensional space. This mathematical result is illustrated via applications to a number of important problems of estimation and robustness under spherical uncertainty. These include parameter estimation, characterization of attainability sets of dynamical systems, and robust stability of affine polynomial families.

Pavel Shcherbakov

Papers

Invariance and nonfragility in the rejection of exogenous disturbances

A probabilistic point of view on peak effects in linear difference equations

Why does Monte Carlo fail to work properly in high-dimensional optimization problems?

Randomization in Robustness, Estimation, and Optimization

Random spherical uncertainty in estimation and robustness