Memory-efficient nonsmooth dynamic optimization using adaptive randomized compression

Tank changeover is a routine process in industry forplacing fueltanks into or out of service. The operation must use inert gas toavoid the flammability zone. However, inert gas consumption shouldbe minimized for economic reasons. This requires dynamic modelingand optimization of the process, as addressed in the present work.A new dynamic optimization problem for minimizing the inert gas consumption,while ensuring fire safety is proposed. As part of the problem constraints,the flammability zone is characterized by disjunctive constraints,which are then converted to a new simple, nonsmooth formula, removingthe need for data regression. This together with the multi-mode flowequations in the model leads to a nonsmooth dynamic optimization problem.To enable reliable solution by gradient-based solvers, the problemis reformulated to a smooth one using sigmoid functions. Case studiesof methane tank purging and filling operations demonstrate that theproposed approach is able to minimize the inert consumption by providingoptimal trajectories of the tank inlet and outlet flow rates, whileensuring the operation remains outside the flammability zone. It isshown that the proposed dynamic optimization can yield significanteconomic benefits as it reduces the nitrogen consumption by abouttwo-third in one of the examples solved.

In this paper we discuss computational issues related to optimal sensor placement in numerical weather prediction (NWP). Specifically we will discuss the application of observability as a metric for sensor placement to an atmospheric flow model and the arising optimization problem. Atmospheric data assimilation is the process of estimating the initial system state based on observations needed in NWP to produce a forecast of future weather conditions. Optimal placement of sensors for data assimilation leading to an improvement in the analysis of the data assimilation and improved forecast quality is of great interest. The traditional definition of observability is not necessarily suitable for NWP applications because of the high dimensions used in NWP. We use the concept of partial observability where the observability of a system is computed on a reduced subspace and is obtained using dynamic optimization. This definition allows for a characterization of the observability of complicated systems. Using partial observability for optimal sensor placement leads to a max-min problem. We use an empirical gramian to reduce this problem into one of eigenvalue optimization. Our focus will be to develop computational methods that are both efficient and scalable. We will leverage tools typically available in data assimilation and introduce tools used in nonsmooth optimization. We will use the shallow water equations as a testbed for our method of optimal sensor placement in four dimensional variational data assimilation.

Nonsmooth Analysis and Geometry The Basic Problem in the Calculus of Variations Verification Functions and Dynamic Programming Optimal Control References.

The optimal control literature is dominated by standard problems in which the system cost functional is expressed in the well-known Bolza form. Such Bolza cost functionals consist of two terms: a Mayer term (which depends solely on the final state reached by the system) and a Lagrange integral term (which depends on the state and control values over the entire time horizon). One limitation with the standard Bolza cost functional is that it does not consider the cost of control changes. Such costs should certainly be considered when designing practical control strategies, as changing the control signal will invariably cause wear and tear on the system's acutators. Accordingly, in this paper, we propose a new optimal control formulation that balances system performance with control variation. The problem is to minimize the total variation of the control signal subject to a guaranteed-cost constraint that ensures an acceptable level of system performance (as measured by a standard Bolza cost functional). We first apply the control parameterization method to approximate this problem by a non-smooth dynamic optimization problem involving a finite number of decision variables. We then devise a novel transformation procedure for converting this non-smooth dynamic optimization problem into a smooth problem that can be solved using gradient-based optimization techniques. The paper concludes with numerical examples in fisheries and container crane control.

In this paper we will discuss the application of observability to the planning of sensor configurations in numerical weather prediction (NWP). The dimensions used in NWP make conventional definitions of observability impractical. For this reason we will rely partial observability which is obtained using dynamic optimization to approximate the observability. Using this metric we will form an optimization problem to select sensor configurations that maximize the partial observability of the dynamical system. This leads to a max–min problem which using an empirical gramian matrix we reduce to an eigenvalue optimization problem. Atmospheric data assimilation is the process of combining prior knowledge with observations to form an estimate of the system state required to produce a forecast of future weather conditions. Optimal sensor configurations leading to improved forecast quality are of interest. Due to the potential size of our intended application we will focus on computational methods that are both efficient and scalable. We will also leverage existing tools used in data assimilation and introduce tools used in nonsmooth optimization.

Dynamic optimization of constrained chemical engineering problems using dynamic programming

Problems of nonsmooth nonconvex dynamic optimization, optimal control (in discrete time), including feedback control, and machine learning are considered from a common point of view. An analogy between controlling discrete dynamical systems and multilayer neural network learning problems with nonsmooth objective functionals and connections is traced. Methods for computing generalized gradients for such systems based on the Hamilton–Pontryagin functions are developed. Gradient (stochastic) algorithms for optimal control and learning are extended to nonconvex nonsmooth dynamic systems.

Neural network for nonsmooth pseudoconvex optimization with general convex constraints

Optimal switching instants for a switched-capacitor DC/DC power converter

Nonsmooth Hessenberg differential-algebraic equations

Minimizing control volatility for nonlinear systems with smooth piecewise-quadratic input signals

Mathematical programming has become a valuable tool in process engineering. However, optimization of hybrid dynamic systems with autonomous mode transitions still constitutes a major challenge for theoretical treatments and engineering application. Among the existing approaches for addressing this obstacle, reformulation strategies appear to be most promising. In this study, a modified smoothing strategy and an extended penalization approach to approximate the non-smooth dynamic optimization problem by a smooth one are presented. As a result, a local solution can be gained by a NLP solver after a discretization of the smoothed problem. This solution converges to that of the original non-smooth problem when the value of the introduced reformulation parameter goes to zero. Heuristic rules to select parameter values for both strategies are proposed based on their inherent features. Results from two case studies indicate the capability of the proposed approaches to efficiently obtain physically meaningful solutions.

A DIRECT METHOD FOR SOLVING NONSMOOTH OPTIMAL CONTROL PROBLEMS

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min-max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.

Problems: local min, numerically complex, lacking derivatives, large-scale.Theory/software exists for smoothed/convex counterparts (IPOPT, CVX, various Matlab/Julia/Python/... implementations).Memory required to store state trajectory (and auxiliary info like Lagrange multipliers) is often prohibitively expensive: O(N(M + m)) for N 10 5 , M 10 10 . Key Algorithmic Requirements:1. Nonconvex functions and nonsmooth regularizers.2. Handle large-scale problems with rapid convergence, mesh independence, and matrix free operations.3. Leverage inexactness by proving convergence for j, j computed inexactly via discretization, reduced-order modeling, compression, etc. * rank 1 experiment failed to converge due to step-size tolerance.The final adapted rank was r = 8 leading to 26x compression.