Polynomial approximation of inequality path constraints in dynamic optimization

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Abstract

We propose an algorithm for dynamic optimization problems with inequality path constraints. It solves a sequence of approximated problems where the path constraint is imposed on a finite number of points. Between adjacent points, an approximating polynomial of the constraint value is calculated and an additional constraint is imposed on the maximum value of this polynomial. We consider Taylor and Hermite polynomials. New points are added based on constraint violations or large approximations errors of the approximating polynomials. We prove finite convergence to a feasible point assuming: (i) the dynamic optimization problem has a Slater point, (ii) pointwise constraints are respected at each iteration. We compare the performance of the algorithm with the algorithm by Fu et at. (Automatica 62, 2015, p. 184–192) for three small case studies and an up-to-date industrial application where we calculate optimal feed rates for a semi-batch emulsion polymerization reactor. The results show that our proposed algorithm needs to solve fewer subproblems, i.e. fewer iterations, at the cost of more constraints, resulting in smaller CPU times.

Introduction

Dynamic optimization problems (DOPs) with inequality path constraints arise in many engineering areas. Some of these problems require the path constraints to be respected rigorously along the entire domain of the independent variable(s).

There are some approaches to solve DOPs with inequality path constraints using indirect methods based on maximum principles, see e.g., the review by Hartl et al. (1995). However, it is difficult to apply indirect methods to complex problems (Biegler, 2007). On the other side, direct transcription methods, where the infinite dimensional problem is transformed into a finite one, typically enforce the constraints only at a finite number of points, thus not guaranteeing the constraints along the entire integration domain. In fact, often some constraint violations are observed.

Some methods have been proposed to enforce the path constraints in the entire domain. Jacobson and Lele (1969), Jacobson et al. (1971) and Bell and Sargent (2000) introduced slack variables to transform inequalities in equalities, and add a penalty term to the objective function. The path constraint can also be transformed into an end-point constraint, by integrating the violation of the constraint along time (Teo, Goh, 1987, Teo, Jennings, 1989, Vassiliadis, Sargent, Pantelides, 1994, Loxton, Teo, Rehbock, Yiu, 2009). Feehery and Barton (1998) proposed the dynamic detection of the regions where the constraint is active, solving the equality constrained problem when the constraint is active and the unconstrained problem when it is not. On the basis of global dynamic optimization, Zhao and Stadtherr (2011) provided a method with guaranteed satisfaction of path constraint improving the algorithm developed by Lin and Stadtherr (2007).

A different class of algorithms guarantee feasible solutions for DOPs by solving a sequence of nonlinear programs (NLPs). These methods update the set of points where the constraints are enforced along the iterations based on the intervals where there were violations in the previous iterations. Among these methods, Chen and Vassiliadis (2005) proposed a dynamic population of the points based on end-points and middle points of the intervals where the constraints are not respected. Their algorithm can be interpreted as an extension of the algorithm of Blankenship and Falk (1976) to dynamic problems. Potschka et al. (2009) proposed a different formulation where the discrete points are selected based on the local minima of the constraint inside the intervals of constant controls. The connection between DOPs with path constraints and semi-infinite programs was explicitly used by Fu et al. (2015) to formulate a new algorithm where the update is based on the time point where a maximum constraint violation if found. The algorithm was developed for dynamic problems with ordinary differential equations (ODEs) embedded, later applied to differential-algebraic equations (DAEs) (Faust et al., 2016), and is an extension of the right-hand restriction method proposed by Mitsos (2011) for semi-infinite programs. The main drawback of this approach, similarly to the one by Chen and Vassiliadis (2005), is the relatively large number of optimization problems solved, since each iteration of the algorithm requires a solution of an NLP.

We propose a new method to solve DOPs by direct methods with guaranteed satisfaction of inequality path constraints and approximated first order optimality conditions. The method first discretizes the path constraint, i.e., imposes it on a finite number of points. This transforms the DOP into an NLP. The main new idea of the method is to use a polynomial approximation of the path constraint in-between the points at which the constraint is enforced. Then, the maximum value of the polynomial is included as a new constraint of the NLP. When the distance between two points is small, the polynomials are a good approximation of the path constraint, and the maximum point of the polynomial approximates the maximum value of the path constraint over the interval. We investigate different alternatives to generate the polynomial function: second-degree polynomial based on Taylor expansion, as well as second and third-degree polynomials based on Hermite interpolation. In order to reduce the approximation error, the set of points where the constraints are evaluated are updated along the iterations. New points are added to reduce the distance in-between the points where violations occur and when the error of the approximation is greater than specified tolerances. The algorithm is proven to terminate finitely under mild assumptions. We test the algorithm on three small case studies and compare against the algorithm proposed by Fu et al. (2015). In the investigated case studies, the algorithm requires a lower number of NLPs; however, the NLPs solved by the algorithm have more inequality constraints than the ones solved by Fu et al. (2015).

In the next section, the DOP with path constraints is stated. Section 4 presents the development of the algorithm for dynamic optimization with inequality path constraints, following by a demonstration of the algorithm in Section 5. Finally, Section 6 concludes the paper and discusses some possible future work.

Section snippets

Problem statement

We are interested in solving DOPs with a finite number of degrees of freedom pP={pIRnp:pa,ipipb,i,i=1,,np}, originating from optimal control problems after control vector parametrization. The states are obtained by integration of a system of ODEs or DAEs. The resulting DOP can be viewed as a semi-infinite program, i.e., an optimization problem with a finite number of degrees of freedom and an infinite number of constraints (Stein, Steuermann, 2012, Fu, Faust, Chachuat, Mitsos, 2015). It is

Summary of semi-infinite program based algorithm (Fu et al., 2015)

A semi-infinite program is an optimization problem with finitely many degrees of freedom and infinitely many constraints. Fu et al. (2015) used the connection of dynamic optimization problems with inequality path constraints and semi-infinite programs to adapt the algorithm proposed by Mitsos (2011) to find guaranteed feasible solutions for these problems with approximated optimality conditions. The algorithm proposed by Fu et al. (2015) consists of solving a sequence of approximated problems

Algorithm

Similarly to Fu et al. (2015), the algorithm solves the DOP by a sequence of NLPs. Each of those NLPs is generated by imposing the constraint only on a finite number of points Td={t1,,ti,ti+1,,tn}T, resulting in a relaxed DOP. Violations of the constraint may thus occur between the points in Td. The main new idea is to additionally approximate the path constraint in the open interval between two neighboring points ti and ti+1 by a polynomial. The algorithm imposes the maximum value of this

Case studies

We apply the algorithm to four optimization problems. Moreover, the DOPs are also solved with the algorithm proposed by Fu et al. (2015) to compare the performance. All the optimizations are implemented in MATLAB Version 9.5.0 (R2018b, win64), using the NLP solver fmincon with the algorithm active-set to solve the NLP, and the solver ode45 to integrate the model over time. The dynamic optimizations are performed on a server with an Intel Xeon CPU E5-2630 v4 @ 2.20 GHz processor, 128 GB RAM,

Conclusion and future work

An algorithm for dynamic optimization of systems described by DOE and DAE with guaranteed satisfaction of inequality path constraints is proposed. The algorithm approximates the path constraint by a polynomial and a constraint on the maximum value is included. The algorithm is proven to provide a local optimal point for an approximate εstat-stationary tolerance with εact-active constraints in a finite number of iterations under mild assumptions.

We investigate different case studies showing that

Acknowlgedgments

This work has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreement no. 675251. The authors thank Dr. Nida Sheibat-Othman for providing the dynamic model of the emulsion polymerization reactor.

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