Bounding the stationary distributions of the chemical master equation via mathematical programming

The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schl¨ogl’s model for a chemical bifurcation, a two-dimensional toggle switch, a model for bursty gene expression, and a dimerisation model with multiple stationary distributions

Combining Top-Down and Bottom-up in Energy Policy Analysis: A Decomposition Approach

The formulation of market equilibrium problems as mixed complementarity problems (MCP) permits integration of bottom-up programming models of the energy system into top-down general equilibrium models of the overall economy. Despite the coherence and logical appeal of the integrated MCP approach, implementation cost and dimensionality both impose limitations on its practical application. A complementarity representation involves both primal and dual relationships, often doubling the number of equations and the scope for error. When an underlying optimization model of the energy system includes upper and lower bounds on many decision variables the MCP formulation may suffer in robustness and efficiency. While bounds can be included in the MCP framework, the treatment of associated income effects is awkward. We present a decomposition of the integrated MCP formulation that permits a convenient combination of top-down general equilibrium models and bottom-up energy system models for energy policy analysis. We advocate the use of complementarity methods to solve the top-down economic equilibrium model and quadratic programming to solve the underlying bottom-up energy supply model. A simple iterative procedure reconciles the equilibrium prices and quantities between both models. We illustrate this approach using a simple stylized model. --Mathematical Programming,Mixed Complementarity,Top-Down/Bottom-Up

New error measures and methods for realizing protein graphs from distance data

The interval Distance Geometry Problem (iDGP) consists in finding a realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver$+$formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates

Active sequential hypothesis testing

Consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration. Due to the sequential nature of the problem, the decision maker relies on his current information state to adaptively select the most ``informative'' sensing action among the available ones. In this paper, using results in dynamic programming, lower bounds for the optimal total cost are established. The lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability. Moreover, upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions. It is shown that the first proposed heuristic achieves asymptotic optimality, where the notion of asymptotic optimality, due to Chernoff, implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration (hence the number of collected samples) increases. The second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search. However, by considering the dependency on the number of hypotheses, under a technical condition, this second heuristic is shown to achieve a nonzero information acquisition rate, establishing a lower bound for the maximum achievable rate and error exponent. In the case of a noisy dynamic search with size-independent noise, the obtained nonzero rate and error exponent are shown to be maximum.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1144 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Linearly Convergent First-Order Algorithms for Semi-definite Programming

In this paper, we consider two formulations for Linear Matrix Inequalities (LMIs) under Slater type constraint qualification assumption, namely, SDP smooth and non-smooth formulations. We also propose two first-order linearly convergent algorithms for solving these formulations. Moreover, we introduce a bundle-level method which converges linearly uniformly for both smooth and non-smooth problems and does not require any smoothness information. The convergence properties of these algorithms are also discussed. Finally, we consider a special case of LMIs, linear system of inequalities, and show that a linearly convergent algorithm can be obtained under a weaker assumption