Erratum: Validated Linear Relaxations and Preprocessing: Some Experiments

This is a correction to R. B. Kearfott and S. Hongthong’s article [SIAM J. Optim., 16 (2005), pp. 418–433]. DOI. 10.1137/100816080 There are errors in column 4 (entitled “Under / over estimators”) of rows 3, 4, and 5 of Table 4.1, on p. 426 of [1]. As a consequence, references to those entries in the table on lines 1 and 2 below the table (on page 426) and on lines 5, 6, 7, and 8 of section 4.1 are incorrect, and the last line of that paragraph should be deleted. The error occurred by using the wrong enclosure range. For example, the midpoint of the enclosure range [−3, 1] for v4, namely v4 = −1, should have been used in the tangent line for row 3, whereas the midpoint of the enclosure range for v5 = v 2 4 had been used instead. Similar errors occurred in rows 4 and 5. Table 4.1 should therefore be corrected to read as follows: Operation Enclosures Under/over estimators Convexity 1 v3 ← x1 + x2 [−2, 2] x1 + x2 − v3 = 0 linear 2 v4 ← v3 − 1 [−3, 1] v3 − 1− v4 = 0 linear 3 v5 ← v2 4 [0, 9] (−1)2 + 2(−1)(v4 − (−1)) − v5 ≤ 0 convex 4 v6 ← x1 [0, 1] (0)2 + 2(0)(v1 − 0)− v6 ≤ 0 convex 1− v6 ≥ 0 nonconvex 5 v7 ← x2 [0, 1] (0)2 + 2(0)(v2 − 0)− v7 ≤ 0 convex 1− v7 ≥ 0 nonconvex 6 v8 ← v6 + v7 [0, 2] v6 + v7 − v8 = 0 linear 7 v9 ← v8 − 1 [−1, 1] v8 − 1− v9 = 0 linear 8 v10 ← −v2 9 [−1, 0] −1− v10 ≤ 0 nonconvex 9 v11 ← v5 + v10 [−1, 9] v5 + v10 − v11 ≤ 0 linear Thus, lines 1 and 2 below Table 4.1 should read as follows: enclosure interval; for example, the expression (−1)2 + 2(−1)(v4 − (−1)) in the third row corresponds to the tangent line to v 4 at v4 = −1. The nonconvex operations (−v2 9 , Since the data were no longer correct, the last sentence of the paragraph above section 4.1 should be deleted. (The reader may solve the corrected linear program with any method.) Similarly, the first paragraph of section 4.1 (in which lines 5, 6, 7, and 8 are changed) should read as follows: 4.1. Refining convex constraints. As explained in [23, section 4.2] and elsewhere, the nonlinear convex operations can be approximated more closely in the linear relaxation by appending more constraints corresponding to additional tangent lines. For example, in the nonlinear convex operation v5 ← v 4 in Example 1, in addition ∗Received by the editors November 30, 2010; accepted for publication (in revised form) December 3, 2010; published electronically March 31, 2011. http://www.siam.org/journals/siopt/21-1/81608.html †Department of Mathematics, University of Louisiana, Lafayette, LA 70504-1010 (rbk@louisiana. edu).