Algorithmic differentiation techniques for global optimization in the COCONUT environment

Abstract

We describe algorithmic differentiation as it can be used in algorithms for global optimization. We focus on the algorithmic differentiation methods implemented in the COCONUT Environment for global nonlinear optimization.The COCONUT Environment represents each factorable optimization problem as a directed acyclic graph (DAG). Various inference modules implemented in this software environment can serve as building blocks for solution algorithms. Many of them use techniques based on various forms of algorithmic differentiation for computing approximations or enclosures of functions or their derivatives.The algorithmic differentiation in the COCONUT Environment does not only provide point evaluations but also range enclosures of derivatives up to order 3, as well as slopes up to second order. Care is taken to ensure that rounding errors are treated correctly. The ranges of the enclosures can be tightened by combining the evaluation routines with constraint propagation. Advantages and pitfalls of this method are also outlined.

Verified Global Optimization for Estimating the Parameters of Nonlinear Models

“…For computing centered forms, etc., COCONUT provides various algorithmic differentiation tools [28]. We computed ∇ f ([x]) by backward algorithmic differentiation.…”

“…The range of the gradient over some [x] can also be enclosed using various methods. It can be computed by forward and backward algorithmic differentiation [28,33], the forward evaluation giving better enclosures but taking an effort of O(n) function evaluations, while the backward method produces slightly worse enclosures but requires an effort of only about two function evaluations. For both approaches the overestimation is O(rad ([x])).…”

Best practices for comparing optimization algorithms

“…A variety of information is reported such as the number of global numerical solutions found (i.e., the best solution found among all optimization algorithms), number of local solutions found, number of wrong claims, etc. For problem representation, it uses Directed Acyclic Graphs (DAGs) from the Coconut Environment [SM12]. This user-friendly environment analyzes results and automatically summarizes them before reporting them in an easy-to-use format such as L A T E X, JPEG, and PDF.…”

Rigorous filtering using linear relaxations

“…The ICOS solver by Lebbah [17] is a free software package for solving nonlinear and continuous constraints, based on constraint programming, relaxation and interval analysis techniques. The prize winning, commercial solver Baron by Sahinidis & Tawarmalani [26] -a highly developed, approximate global optimization solver -uses a special linear relaxation technique called the sandwich method, while the COCONUT Environment [27,28] applies both linear relaxations using slopes and reformulation-linearization on Directed Acyclic Graphs (DAGs). Note that solving constrained global optimization problems by branch and bound is in practice reduced to solving a sequence of constraint satisfaction problems, each obtained by adding a constraint f (x) ≤ f best to the original constraints, where f is the objective function and f best the function value of the best feasible point found so far.…”