The lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks. To deal with this problem, we propose a method for the analysis of genetic regulatory networks under parameter uncertainty. We consider models based on piecewise-multiaffine differential equations, dynamical properties expressed in temporal logic, and intervals for the values of uncertain parameters. The problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value – the corresponding parameter set is then called valid – or to find valid subsets of a given parameter set. The proposed method uses discrete abstractions and model checking, and allows for efficient search of the parameter space. However, the abstraction process creates spurious behaviors in the abstract systems, along which time does not progress. Consequently, the verification of liveness properties, expressing that something will eventually happen, and implicitly assuming progress of time, often fails. A solution to this second problem is proposed using the notion of transient regions. This approach has been implemented in a tool for robust verification of gene networks (RoVerGeNe) and applied to the tuning of a synthetic network built in E. coli.

Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs with a parameter in the cost, or equivalently the right-hand side of the constraints, and solved explicitly offline. Degeneracy occurs when the control input, or optimiser, is non-unique, which can cause chattering of the control input and overlap of the polyhedral regions of the explicit solution. This paper introduces a new and efficient approach to the computation of the solution to a multiparametric linear program with the parameter in the cost in the presence of degeneracy. Rather than solve the degenerate problem directly, we solve a lexicographically (symbolically) perturbed version of it that is guaranteed to be non-degenerate. We show that every optimal solution of the perturbed problem is an optimal solution to the original and that the perturbed solution is continuous, unique and defined over a set of non-overlapping polyhedral regions. Furthermore, we introduce a new method for computing the optimal solution in an adjacent region that is very efficient in all cases, and reduces to a single simplex pivot for non-degenerate regions. The proposed algorithm is particularly suited for the calculation of the explicit solution to a class of constrained optimal control problems, since it ensures that the control input is everywhere continuous and unique, thereby removing the danger of chattering in problems with linear costs. The algorithm is compared through example to existing proposals and a significant complexity improvement is demonstrated.

In this paper we state a generalized form of the policy improvement algorithm for reinforcement learning. This new algorithm can be used to find stochastic policies that optimize single-agent behavior for several environments and reinforcement functions simultaneously. We first introduce a geometric interpretation of policy improvement, define a framework to apply one policy to several environments, and propose the notion of balanced policies. Finally we explain the algorithm and present examples.

disturbances

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”-case of CompletenessC has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).

This article addresses the problem of the computation of instantaneous firing speed in Invariant Behavior state (IB-state) of Constant speed Continuous Petri Net (CCPN) with presence of actual conflicts. The adopted approach is based on polyhedral computations applied to specify an area of possible instantaneous firing speed. If the actual conflicts are resolved by global priorities, the instantaneous firing speed is found in a set of the polytop vertices or alternatively it is found by one formulation of the linear programming problem per each priority level. The approach shown in this article assumes the speed maximisation being prior to priority resolution.

One fundamental problem in the field of network coding is to determine the network coding capacity of networks under various network coding schemes. In this thesis, we address the problem with two approaches: matroidal networks and capacity regions. In our matroidal approach, we prove the converse of the theorem which states that, if a network is scalar-linearly solvable then it is a matroidal network associated with a representable matroid over a finite field. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields in the framework of matroidal networks. We prove a theorem about the scalar-linear solvability of networks and field characteristics. We provide a method for generating scalar-linearly solvable networks that are potentially different from

In this paper we state a generalized form of the policy improvement algorithm for reinforcement learning. This new algorithm can be used to ...nd stochastic policies that optimize single-agent behavior for several environments and reinforcement functions simultaneously. We ...rst introduce a geometric interpretation of policy improvement, de...ne a framework to apply one policy to several environments, and propose the notion of balanced policies. Finally we explain the algorithm and present examples. 1 Idea Until now reinforcement learning has been applied to abstract behavior within one environment. Several methods to ...nd optimal policies for one environment are known [1], [5], [9]. In our research we focus on a general point of view of behavior that appears independently from a single environment. As an example imagine that a robot should learn to avoid obstacles. What we have in mind is a behavior suitable for several environments. Obviously a policy for several environments can not - in general - be optimal for each one of them. Improving a policy for one environment may result in a worse performance in an other. Nevertheless it is often possible to improve a policy for several environments. Finding an "optimal" policy for several environments is related to game theory. In this context our task would be to ...nd one strategy for a game that performs well against several players who follow ...xed strategies. We also consider a similar notion to that of an equilibrium point in game theory [7]. Compared to multiagent reinforcement learning [2], [6], where several agents act in one environment, we have one agent acting in several environments. 2 Notation We propose the following notation for reinforcement learning models, which allows us to treat several environme...

We present a new method for robust dynamic programming that is suitable for linearly constrained polytopic systems with piecewise linear cost functions. The method avoids Bellman's "curse of dimensionality " and performs exact dynamic programming without any discretization of the state space by using polyhedral representations of the cost-to-go functions and feasible sets. It is shown that the robust dynamic programming recursion preserves this polyhedral representation. It is explained in detail how these computations can be performed efficiently. We show how the method can be applied to an example problem, namely a car with uncertain mass that shall quickly park in front of a wall without hitting it.

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