For a continuous-time phase-type (PH) distribution, starting with its Laplace–Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

We develop and analyze an unexplored mechanism to reduce biorefinery supply chain costs when the feedstock is a perennial crop: adjusting the age structure, and hence yield, of the perennial feedstock. The non-monotonicity of the age-yield function introduces a non-convexity to the cost minimization problem. We show that, despite this, the problem has a solution and present analytic and numeric comparative statics, finding that larger refineries are most likely to benefit from optimizing age structure. The model is calibrated to the sugarcane industry in Brazil. The cost reductions from optimizing age, compared to the observed regional average age, are less than 1%.

We consider the problem of providing optimal uncertainty quantification (UQ) – and hencerigorous certification – for partially-observed functions. We present a UQ frameworkwithin which the observations may be small or large in number, and need not carryinformation about the probability distribution of the system in operation. The UQobjectives are posed as optimization problems, the solutions of which are optimal boundson the quantities of interest; we consider two typical settings, namely parametersensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. Thesolutions of these optimization problems depend non-trivially (even non-monotonically anddiscontinuously) upon the specified legacy data. Furthermore, the extreme values are oftendetermined by only a few members of the data set; in our principal physically-motivatedexample, the bounds are determined by just 2 out of 32 data points, and the remaindercarry no information and could be neglected without changing the final answer. We proposean analogue of the simplex algorithm from linear programming that uses these observationsto offer efficient and rigorous UQ for high-dimensional systems with high-cardinalitylegacy data. These findings suggest natural methods for selecting optimal (maximallyinformative) next experiments.