Abstract This article proposes new practical methods for furnishing generalized derivative information of optimal-value functions with embedded parameterized convex programs, with potential applications in nonsmooth equation-solving and optimization. We consider three cases of parameterized convex programs: (1) partial convexity—functions in the convex programs are convex with respect to decision variables for fixed values of parameters, (2) joint convexity—the functions are convex with respect to both decision variables and parameters, and (3) linear programs where the parameters appear in the objective function. These new methods calculate an LD-derivative, which is a recently established useful generalized derivative concept, by constructing and solving a sequence of auxiliary linear programs. In the general partial convexity case, our new method requires that the strong Slater conditions are satisfied for the embedded convex program’s decision space, and requires that the convex program has a unique optimal solution. It is shown that these conditions are essentially less stringent than the regularity conditions required by certain established methods, and our new method is at the same time computationally preferable over these methods. In the joint convexity case, the uniqueness requirement of an optimal solution is further relaxed, and to our knowledge, there is no established method for computing generalized derivatives prior to this work. In the linear program case, both the Slater conditions and the uniqueness of an optimal solution are not required by our new method.

BACKGROUND: Dynamic Flux Balance Analysis (DFBA) is a dynamic simulation framework for biochemical processes. DFBA can be performed using different approaches such as static optimization (SOA), dynamic optimization (DOA), and direct approaches (DA). Few existing simulators address the theoretical and practical challenges of nonunique exchange fluxes or infeasible linear programs (LPs). Both are common sources of failure and inefficiencies for these simulators. RESULTS: DFBAlab, a MATLAB-based simulator that uses the LP feasibility problem to obtain an extended system and lexicographic optimization to yield unique exchange fluxes, is presented. DFBAlab is able to simulate complex dynamic cultures with multiple species rapidly and reliably, including differential-algebraic equation (DAE) systems. In addition, DFBAlab's running time scales linearly with the number of species models. Three examples are presented where the performance of COBRA, DyMMM and DFBAlab are compared. CONCLUSIONS: Lexicographic optimization is used to determine unique exchange fluxes which are necessary for a well-defined dynamic system. DFBAlab does not fail during numerical integration due to infeasible LPs. The extended system obtained through the LP feasibility problem in DFBAlab provides a penalty function that can be used in optimization algorithms.

Bundle methods for nonsmooth optimization and semismooth Newton methods for nonsmooth equation solving both require computation of elements of the (Clarke) generalized Jacobian, which provides slope information for locally Lipschitz continuous functions. Since the generalized Jacobian does not obey sharp calculus rules, this computation can be difficult. In this article, methods are developed for evaluating generalized Jacobian elements for a nonsmooth function that is expressed as a finite composition of known elemental piecewise differentiable functions. In principle, these elemental functions can include any piecewise differentiable function whose analytical directional derivatives are known. The methods are fully automatable, and are shown to be computationally tractable relative to the cost of a function evaluation. An implementation developed in C++ is discussed, and the methods are applied to several example problems for illustration.

Processes that repeat in time, such as the cell cycle, the circadian rhythm, and seasonal variations, are prevalent in biology. Mathematical models can represent our knowledge of the underlying mechanisms, and numerical methods can then facilitate analysis, which forms the foundation for a more integrated understanding as well as for design and intervention. Here, the intracellular molecular network responsible for the mammalian circadian clock system was studied. A new formulation of detailed sensitivity analysis is introduced and applied to elucidate the influence of individual rate processes, represented through their parameters, on network functional characteristics. One of four negative feedback loops in the model, the Per2 loop, was uniquely identified as most responsible for setting the period of oscillation; none of the other feedback loops were found to play as substantial a role. The analysis further suggested that the activity of the kinases CK1delta and CK1varepsilon were well placed within the network such that they could be instrumental in implementing short-term adjustments to the period in the circadian clock system. The numerical results reported here are supported by previously published experimental data.

Dynamic optimization problems with linear hybrid (discrete/continuous) systems embedded whose transition times vary are inherently nonconvex. For a wide variety of applications, a certificate of global optimality is essential, but this cannot be obtained using conventional numerical methods. We present a deterministic framework for the solution of such problems in the continuous time domain. First, the control parametrization enhancing transform is used to transform the embedded dynamic system from a linear hybrid system with scaled discontinuities and varying transition times into a nonlinear hybrid system with stationary discontinuities and fixed transition times. Next, a recently developed convexity theory is applied to construct a convex relaxation of the original nonconvex problem. This allows the problem to be solved in a branch-and-bound framework that can guarantee the global solution within epsilon optimality in a finite number of iterations.