Tobias Sutter

Performance Bounds for the Scenario Approach and an Extension to a Class of Non-Convex Programs

We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the optimal value of SCP to the optimal values of RCP and CCP in which the uncertainty takes values in a general, possibly infinite dimensional, metric space. We then extend our results to a certain class of non-convex problems that includes, for example, binary decision variables. In the process, we also settle a measurability issue for a general class of scenario programs, which to date has been addressed by an assumption. Finally, we demonstrate the applicability of our results on a benchmark problem and a problem in fault detection and isolation.

Approximation of constrained average cost Marks

This paper considers discrete-time constrained Markov control processes (MCPs) under the long-run expected average cost optimality criterion. For Borel state and action spaces a two-step method is presented to numerically approximate the optimal value of this constrained MCPs. The proposed method employs the infinite-dimensional linear programming (LP) representation of the constrained MCPs. In particular, we establish a bridge from the infinite-dimensional LP characterization to a finite LP consisting of a first asymptotic step and a second step that provides explicit bounds on the approximation error. Finally, the applicability and performance of the theoretical results are demonstrated on an LQG example.

Efficient Approximation of Quantum Channel Capacities

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M3 log(N)1/2ε-1) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.

Efficient Approximation of Channel Capacities

We propose an iterative method for approximately computing the capacity of discrete memoryless channels, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. The presented method requires O(M2 N√log N/ε) to provide an estimate of the capacity to within ε, where N and M denote the input and output alphabet size; a single iteration has a complexity O(MN). We also show how to approximately compute the capacity of memoryless channels having a bounded continuous input alphabet and a countable output alphabet under some mild assumptions on the decay rate of the channels tail. It is shown that discrete-time Poisson channels fall into this problem class. As an example, we compute sharp upper and lower bounds for the capacity of a discrete-time Poisson channel with a peak-power input constraint.

Efficient approximation of discrete memoryless channel capacities

We propose an iterative method for efficiently approximating the capacity of discrete memoryless channels, possibly having additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To find an ε-approximation of the capacity, in case of no additional input constraints, the presented method has a computational complexity O(1 over εM2N√(logN)), where N and M denote the input and output alphabet size, and a single iteration has a complexity O(MN).

From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first-order methods, leading to a priori as well as a posteriori performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems in the context of Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a fisheries management problem.

On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems

We revisit the linear programming approach to deterministic, continuous time, infinite horizon discounted optimal control problems. In the first part, we relax the original problem to an infinite-dimensional linear program over a measure space and prove equivalence of the two formulations under mild assumptions, significantly weaker than those found in the literature until now. The proof is based on duality theory and mollification techniques for constructing approximate smooth subsolutions to the associated Hamilton–Jacobi–Bellman equation. In the second part, we assume polynomial data and use Lasserre’s hierarchy of primal-dual moment-sum-of-squares semidefinite relaxations to approximate the value function and design an approximate optimal feedback controller. We conclude with an illustrative example.

Capacity approximation of memoryless channels with countable output alphabets

We present a new algorithm, based on duality of convex programming and the specific structure of the channel capacity problem, to iteratively construct upper and lower bounds for the capacity of memoryless channels having continuous input and countable output alphabets. Under a mild assumption on the decay rate of the channels tail, explicit bounds for the approximation error are provided. We demonstrate the applicability of our result on the discrete-time Poisson channel having a peak-power input constraint.

Isospectral flows on a class of finite-dimensional Jacobi matrices☆

Abstract We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes n × n zero-diagonal Jacobi matrices employed as its initial condition. This o.d.e. features a right-hand side with a nested commutator of matrices and structurally resembles the double-bracket o.d.e. studied by R.W. Brockett in 1991. We prove that its solutions converge asymptotically, that the limit is block-diagonal, and above all, that the limit matrix is defined uniquely as follows: for n even, a block-diagonal matrix containing 2 × 2 blocks, such that the super-diagonal entries are sorted by strictly increasing absolute value. Furthermore, the off-diagonal entries in these 2 × 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition. For n odd, there is one additional 1 × 1 block containing a zero that is the top left entry of the limit matrix. The results presented here extend some early work of Kac and van Moerbeke.

Convex programming in optimal control and information theory.

The main theme of this thesis is the development of computational methods for classes of infinite-dimensional optimization problems arising in optimal control and information theory. The first part of the thesis is concerned with the optimal control of discrete-time continuous space Markov decision processes (MDP). The second part is centred around two fundamental problems in information theory that can be expressed as optimization problems: the channel capacity problem as well as the entropy maximization subject to moment constraints.