Abstract
In deterministic continuous constrained global optimization, upper bounding the objective function generally resorts to local minimization at several nodes/iterations of the branch and bound. We propose in this paper an alternative approach when the constraints are inequalities and the feasible space has a non-null volume. First, we extract an inner region, i.e., an entirely feasible convex polyhedron or box in which all points satisfy the constraints. Second, we select a point inside the extracted inner region and update the upper bound with its cost. We describe in this paper two original inner region extraction algorithms implemented in our interval B&B called IbexOpt (AAAI, pp 99–104, 2011). They apply to nonconvex constraints involving mathematical operators like , \( +\; \bullet ,\; /,\; power,\; sqrt,\; exp,\; log,\; sin\). This upper bounding shows very good performance obtained on medium-sized systems proposed in the COCONUT suite.
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Notes
We consider minimization in this paper without loss of generality.
An interval \([x_i]=[\underline{x_i},\overline{x_i}]\) defines the set of reals \(x_i\) s.t. \(\underline{x_i} \le x_i \le \overline{x_i}\). A box \([x]\) is the Cartesian product of intervals \([x_1] \times \cdots \times [x_i] \times \cdots \times [x_n]\).
\(\epsilon \)-minimize \(f(x)\) means minimize \(f(x)\) with a precision \(\epsilon \), i.e., we have \(f(y) \ge f(x)-\epsilon \), for all (feasible) \(y\).
In addition, Chabert and Beldiceanu handled a dual problem consisting in finding a box with no solution (i.e., all points in this box violate the constraints) and required an initial point to be “inflated” to a box...
With floating-point numbers, an interval evaluation is conservative (i.e., contains all the real-valued images) if the lower bound of the interval image is rounded towards \(-\infty \) while the upper bound is rounded to \(+\infty \). Both rounding operations constitute a so-called outward rounding. With inward rounding, the lower bound is rounded towards \(+\infty \) and the upper bound towards \(-\infty \). The property enforced by outward (resp. inward) rounding for an operator \(op\) is (1), resp. (2):
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1.
\(\forall x \in [x] ~~ \exists z \in op([x]) ~~ z=op(x)\)
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2.
\(\forall z\in op([x]) ~~ \exists x \in [x] ~~ z=op(x)\)
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1.
In the case 2, only one inner interval is considered among the different ones in the union. In the case 3, only one maximal inner box is computed (using a random choice on \(x_1\)) among an infinite number of possibles boxes. The last case gathers both drawbacks (from cases 2 and 3). Consider for instance the case of the multiplication \(x_1 \cdot x_2 \in [z]\) where \(z\) is positive (see left side of Fig. 6).
The difficulty is only related to our implementation. The maximality can be more easily checked with a direct implementation.
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Ignacio Araya is supported by the Fondecyt Project 11121366 and the UTFSM Researcher Associated Program.
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Araya, I., Trombettoni, G., Neveu, B. et al. Upper bounding in inner regions for global optimization under inequality constraints. J Glob Optim 60, 145–164 (2014). https://doi.org/10.1007/s10898-014-0145-7
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DOI: https://doi.org/10.1007/s10898-014-0145-7