Numerical methods for non-smooth equation-solving and optimization often require generalized derivative information in the form of elements Clarke Jacobian or B-subdifferential. It is shown here that piecewise differentiable functions are lexicographically smooth sense Nesterov, lexicographic derivatives these comprise a particular subset both B-subdifferential Jacobian. Several recently developed evaluation composite to produce identical results, which also derivatives. A vector forward...

A new model formulation and solution strategy for the design simulation of processes involving multistream heat exchangers (MHEXs) is presented. The approach combines an extension pinch analysis with explicit dependence on exchange area in a nonsmooth equation system to create which solves up three unknown variables MHEX. Recent advances automatic generation derivative‐like information equations make method tractable, use solving methods very precise. Several illustrative examples case study...

A new method for evaluating generalized derivatives in nonsmooth problems is reviewed. Lexicographic directional (LD-)derivatives are a recently developed tool analysis derivative elements tractable and robust way. Applicable to both steady-state dynamic settings, LD-derivatives exhibit number of advantages over current theory algorithms. As highlighted this article, the LD-derivative approach now admits suitable inverse implicit functions, dynamical systems optimization problems, among...

Automatic generation of convex relaxations and subgradients is critical in global optimization, typically carried out using variants automatic/algorithmic differentiation (AD). At previous AD conferences, the forward reverse modes were presented to evaluate accurate for supplied composite functions. In a recent approach generating implicit functions, these are constructed as optimal-value functions; this formulation versatile but complicates sensitivity analysis. We present first subgradient...

Nonsmooth functions have been used to model discrete-continuous phenomena such as contact mechanics, and are also prevalent in neural network formulations via activation ReLU. At previous AD conferences, Griewank et al. showed that nonsmooth may be approximated well by piecewise-affine constructed using an AD-like procedure. Moreover, a function always represented "abs-normal form", encoding it collection of four matrices two vectors. We present new general complementarity for root-finding...

A novel subgradient evaluation method is proposed for nonsmooth convex relaxations of parametric solutions ordinary differential equations (ODEs) arising in global dynamic optimization, assuming that the always lie strictly within interval bounds during integration. We argue this assumption reasonable practice. These subgradients are computed as unique solution an auxiliary affine ODE, analogous to classical forward/tangent sensitivity methods smooth systems. Unlike established approaches...

Bundle methods for nonsmooth optimization and semismooth Newton equation solving both require computation of elements the (Clarke) generalized Jacobian, which provides slope information locally Lipschitz continuous functions. Since Jacobian does not obey sharp calculus rules, this can be difficult. In article, are developed evaluating a function that is expressed as finite composition known elemental piecewise differentiable principle, these functions include any whose analytical directional...

Established sensitivity results for hybrid discrete/continuous dynamic systems are generalized by relaxing smoothness assumptions on the functions governing systems' continuous evolution and discrete event handling. The new only require L-smoothness of these in sense Nesterov, instead differentiability. Parametric lexicographic derivatives such a system provide useful local first-order information, described as unique solutions auxiliary systems. This analysis framework permits derivative...

This paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes the active set. is accomplished using lexicographic directional derivatives, a recently developed tool nonsmooth analysis based on Nesterov's differentiation. A implicit function theorem augmented with generalized derivative information and applied standard reformulation of KKT system. It shown that sufficient conditions this variant are implied by point satisfying...

A recent nonsmooth vector forward mode of algorithmic differentiation (AD) computes Nesterov's L-derivatives for composite functions; these provide useful sensitivity information to methods optimization and equation solving. The established reverse AD evaluates gradients efficiently smooth functions, but it does not extend directly functions. Thus, this article examines branch-locking strategies harness the benefits techniques even in case, order improve computational performance mode. In...

We develop a manifold sampling algorithm for the minimization of nonsmooth composite function $f \triangleq \psi + h \circ F$ when $\psi$ is smooth with known derivatives, $h$ known, nonsmooth, piecewise linear function, and $F$ but expensive to evaluate. The trust-region classifies points in domain as belonging different manifolds uses this knowledge computing search directions. Since classifying objective using only values simple. prove that all cluster sequence iterates are Clarke...

Convex relaxations of functions are used to provide bounding information deterministic global optimization methods for nonconvex systems. To be useful, these must converge rapidly the original system as considered domain shrinks. This article examines convergence rates convex outer approximations and nonlinear programs (NLPs), constructed using affine subtangents an existing relaxation scheme. It is shown that inherit rapid second-order pointwise from scheme under certain assumptions....

Recent advances in nonsmooth sensitivity analysis are extended to describe particular elements of Clarke's generalized gradient for the objective function a optimal control problem, terms states an auxiliary dynamic system. The considered problem is generic nonlinear open-loop which cost and right-hand side describing system dynamics may each be nonsmooth. desired obtained under two parametric discretizations function: representation as linear combination basis functions, piecewise constant...

This paper presents the effectiveness of interior search algorithm in economic power scheduling a grid-tied DC microgrid with renewable generation (wind and photovoltaic) battery energy storage. The study modelling simulation various DC/DC converters for tracking maximum from wind photovoltaic sources bidirectional flow its controllers were modelled simulated MATLAB/Simulink. generating units dispatched economically using objective to minimize operating cost microgrid. results verify as...

A theory is developed for local, first-order sensitivity analysis of limit-cycle oscillating hybrid systems, which are dynamical systems exhibiting both continuous-state and discrete-state dynamics whose state trajectories closed, isolated, time-periodic. Methods the computation initial-condition sensitivities parametric to account exactly any jumps in at discrete transitions exploit time-periodicity system. It shown that system can be represented as sum a time-decaying component...