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... subregions that may have good solutions while dropping unpromising subregions. In order to verify the existence of feasible points for a given subregion, they usually utilize interval Newton methods =-=[116, 115]-=-. GlobSol [80] and ILOG solver [108] are popular software packages that implement these methods. Interval methods, however, are unable to cope with general nonlinear constraints and may be misled by i...

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...degeneration. Hansen used this technique in order to prove existence of a feasible point for nonlinear equations within a bounded box. It was further modified and investigated numerically by Kearfott =-=[11]-=-, [12], and is also described in his book [13]. Corresponding algorithms are implemented in his software package GlobSol. 5. Certificate of Infeasibility. In branch and bound algorithms a subproblem i...

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...more complicated than evaluating the objective function at a point. In particular, the point of evaluation must be known to be feasible. We explain the process in GlobSol for verifying feasibility in =-=[18]-=- ands[16, §5.2.4]. This process is presently a weakness that prevents some equalityconstrained problems from being handled as efficiently as unconstrained or certain inequality-constrained problems. G...

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...nd whose functions are not di#erentiable. Hence, methods that require di#erentiability cannot be applied. Such methods include interval methods requiring derivatives (such as interval-Newton methods) =-=[101, 100, 87]-=-, gradient-descent methods, Newton's method [109, 121, 129], trajectory methods [158, 155, 58, 135, 30, 143, 149, 28, 152], covering methods requiring derivatives [38, 39], penalty methods requiring d...

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...mate optimizer ˇx. However, because of likely singularity of this system, this technique is likely to fail; see the discussion in [5, §2] to gain insight into the case where the NLP (1) is linear. In =-=[3]-=-, we proposed and provided test results for a technique for perturbing the approximate optimizing point ˇx and by constructing a box about the perturbed feasible point within which an interval Newton ...

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...ntaining an approximate minimizer. However, in constrained problems such as Problem 1, a rigorous upper bound is obtained with this process only if it is certain that X contains a feasible point (See =-=[5]-=-.) Generally, if a good approximation to a feasible point (or solution of the constraints) is known, interval Newton methods can prove that an actual feasible point exists within specified bounds, at ...

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... that Z(ν, x, y) := F ′(x)T (νa−BT y) ∈ z, (14) and the complementarity conditions max((BFj(x)− bj)yj , (BFj(x)− bj)yj) = 0 (15) hold for j = 1, . . . ,m. These conditions comprise the Karush-John optimality conditions for the problem (8); cf. the derivation and discussion of the history in Schichl and Neumaier [31]. If ν 6= 0 we may rescale the multipliers to have ν = 1, leading to the KuhnTucker optimality conditions. We normalize instead by rescaling so that max(ν, ‖y‖∞) = 1, which is possible even when ν = 0 and leads to bounded multipliers. This is achieved by ν ← 1max(1, ‖y‖∞) , y ← νy. (16) The following result suggests a way to define Lagrange multipliers y ∈ Rm (for the constraints) and ν ∈ R (for the objective) at an arbitrary point x (intended to be an approximate local minimizer). Theorem 1 If, for some x ∈ Rn, the constrained optimization problem min g(y) := ‖µ(Z(1, x, y)− z)‖22 s.t. y ∈ y (17) (with Z from (14) and µ from (6)) has a solution y with g(y) = 0 then (x, y) satisfies the Kuhn-Tucker conditions for (8), and (16) defines associated normalized multipliers satisfying the Karush-John conditions. Proof From g(y) = 0 follows that µ(Z(1, x, y) − z) = 0 implyin...

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... found, then epsilon-inflation is used to construct a small box x, ˜x ∈ x in a subspace of R n within which an interval Newton method can prove there exists a true feasible point; see [5, §5.2.4] and =-=[6]-=-. Using default configuration and initial box x = (a1, a2, a3, x1, x2, x3) = ([−2, 2], [−3, 3], [−3, 3], [−2, 2], [−2, 2], [−2, 2]), the May 2, 2000 version of GlobSol did not complete after processin...