Global dynamic optimization using edge-concave underestimator

Abstract

Optimization of problems with embedded system of ordinary differential equations (ODEs) is challenging and the difficulty is amplified due to the presence of nonconvexity. In this article, a deterministic global optimization method is presented for systems consisting of an objective function and constraints with integral terms and an embedded set of nonlinear parametric ODEs. The method is based on a branch-and-bound algorithm that uses a new class of underestimators recently proposed by Hasan (J Glob Optim 71:735–752, 2018). At each node of the branch-and-bound tree, instead of using a convex relaxation, an edge-concave underestimator or the linear facets of its convex envelope is used to compute a lower bound. The underestimator is constructed by finding valid upper bounds on the diagonal elements of the Hessian matrix of the nonconvex terms. Time dependent bounds on the state variables and diagonal elements of the Hessian are obtained by solving an auxiliary set of ODEs that is derived using the notion of differential inequalities. The performance of the edge-concave relaxation is compared to other approaches on several test problems.

Appendices

Appendix

Continuity and differentiability results of the solution of parametric ordinary differential equations

Consider the following system of ODEs:

$$\begin{aligned} \begin{aligned}& {\dot{y}} = \phi (t,y,x), \\& y(x,t_0) = y_0(x). \end{aligned} \end{aligned}$$

(31)

Let $\varPi$ be a certain domain of $n+p+1$-dimensional space of the variables x, y and t, where $\phi$ is defined. Further, assume that the function $\phi$ and its partial derivatives with respect to y ($\frac{\partial \phi }{\partial y}$) are continuous in $\varPi$.

Theorem 3

[46] If ($t_0$, $x_0$ and $y_0$) is a point in $\varPi$, positive real numbers $\beta _1$ and $\beta _2$ exists such that for

$$\begin{aligned} \begin{aligned}&|x-x_0| < \beta _1, \end{aligned} \end{aligned}$$

the solution

$$\begin{aligned} \begin{aligned}& y = \chi (x,t) \end{aligned} \end{aligned}$$

of Eq. (31), which satisfies the initial condition $\chi (x,t_0)=y_0$ is defined on the interval $|t-t_0| < \beta _2$ and is a continuous function of the variables t and x.

Theorem 4

[46] Let $\frac{\partial \phi }{\partial x}$ exist and be continuous in $\varPi$. If ($t_0$, $x_0$, $p_0$) is a point in $\varPi$, then there exists positive real numbers $\beta '_1$ and $\beta '_2$ such that for

$$\begin{aligned} \begin{aligned}&|x-x_0|< \beta '_1 \text { and } |t-t_0| < \beta '_2, \end{aligned} \end{aligned}$$

the solution of Eq. (31), $\chi (x,t)$ that satisfies the initial condition $\chi (x,t_0)=y_0$, has continuous partial derivative with respect to x i.e., $\frac{\partial \chi (t,x)}{\partial x}$ is continuous.

Corollary 1

[46] Let all the partial derivatives of $\phi$ with respect to the variables x and y exist up to the k-th order inclusive and are continuous, then the solution of Eq. (31), $\chi (x,t)$ also have partial derivatives with respect to the parameters x up to the k-th order inclusive that are continuous.