p. 5 First line of Section 1.4. Change nonzero vector v R n to nonzero vector v C n int([x, y]) = (x, y)

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1 Corrections for the first printing of INTRODUCTION TO NON- LINEAR OPTIMIZATION: THEORY, ALGORITHMS, AND AP- PLICATIONS WITH MATLAB, SIAM, 204, by Amir Beck 2 Last Updated: June 8, 207 p. 5 First line of Section.4. Change nonzero vecr v R n nonzero vecr v C n p. 7 In Example.5, remove the line (it is incorrect when n > ) int([x, y]) = (x, y) p. 7 One line after Definition.6, remove open line segments. p. 8 At the end of the page, the sentence starting from For any i =, 2,..., n, the directional derivative at... until the end of the page should change For any i =, 2,..., n, if the limit f f(x + te i ) f(x) (x) = lim x i t 0 t exists, then it is called the i-th partial derivative of f at x. p.9. Line -9. Replace o( ) : R n + R with o( ) : R + R p.9 two lines before Theorem 2.7, replace charachterization with characterization p. 2 Change A 2 A 2 = UEU T UEU T = UEEU T = UDU = A. A 2 A 2 = UEU T UEU T = UEEU T = UDU T = A. (add transpose the last U) p. 37 Two lines before (3.) change if A if not of full column if A is not of full column p. 47 Line 4. Change (ˆx, R) (ˆx, ˆR) p. 47 In Lemma 3.5, change r = ˆx 2 y n+ ˆr = ˆx 2 y n+ The corrections were incorporated in the second printing of the book 2 I would like thank Heinz Bauschke, David Cohen, Christian Kanzow, Luba Tetruashvili and Yakov Vaisbourd for their invaluable comments that helped compile the list of corrections.

2 p. 58 At the end, replace Therefore, with It holds that p. 62 Last sentence before the MATLAB code. Change p. 68 Change anchor point anchor points p. 69 Two lines before (4.4), change x a, a 2,..., a m x 0 a, a 2,..., a m. p. 72 Change cannot be verified easily cannot be easily guaranteed p. 72 The two occurrences of min{f(x 0 ), f(x ),..., f(x p )} should be replaced with min{f(x 0 ), f(x ),..., f(x m )} p. 73 Line 5 in Section After meaning that add there exists L > 0 for which p. 73 After denote the class by C,. add the following sentence: For a given D R n, the set of all continuously differentiable functions over D whose gradient satisfies the above Lipschitz condition for any x, y D is denoted by C, L (D). p. 74 Lemma The sentence Let f C, L (Rn ). Then for any x, y R n should be replaced with Let D R n and f C, L (D) for some L > 0. Then for any x, y D satisfying [x, y] D it holds that p. 85 Replace the sentence Combining the latter equality with the fact that 2 f(x k ) mi implies that 2 f(x k ) m. Hence, Since 2 f(x k ) mi, it follows that 2 f(x k ) m. Hence, p. 94 Two occurrences of 7 iterations should be replaced with 8 iterations p. 05 Line 3. Change The set C is clearly a convex cone The set C is clearly a convex set p. 05 Example 6.7. Change Lorenz Lorentz p. 05 Last line. Change K 2 = {(x, x 2 ) T : x t + x 2 0} = {(x, x 2 ) : x = 0, x 2 0}, K 2 = {(x, x 2 ) T : x t+x 2 0 for all t R} = {(x, x 2 ) : x = 0, x 2 0}, p. 06 Just before Theorem 6.23, change recall that in Carathéodory n + vecrs recall that in Carathéodory s theorem n + vecrs p. 07 2nd line. Change convex hull conic hull. 2

3 p. In definition 6.33, change Let S R n Let S R n be a convex set p. 8 Line 4. After concave add over a convex set C R n. p. 20 Last line. Replace the global minimizer with a global minimizer. p. 23 Second line in the premise of Theorem 7.2. Add over C between convex and if and only if p. 33 Change Then by the Krein-Milman theorem (Theorem 6.35), for any Then obviously, for any (it is correct as it is, but there s no real need use Krein-Milman) p. 40 Second line of the proof of Lemma It should be y q and not y p p. 47 Second line in Section 8. and one line after (8.), change closed and convex set convex set p. 47 Change level sets of convex sets level sets of convex functions p. 47 Line -8. Delete the sentence starting with The set C is closed and ending with see Theorem p. 49 Change we assume that α i β i+ for any i =, 2,..., n we assume that α i β j for any j > i p. 50 Change since the condition α i β i+ will guarantee since the feasibility condition will guarantee p. 57 Line -4. Should be on R n + instead of on R n p. 69 Line -3. Change local optimum local minimum. p. 7 Line -3. Change differetiable differentiable p. 72 Just before (9.6), change stationry stationary p. 74 Theorem 9.8. Change Then z = P C (x) if and only if Then z = P C (x) if and only if z C and p. 77 In the premise of Lemma 9.2, change Suppose that L L 2 Suppose that L L 2 > 0. p. 82 Equation (9.29) should end with a period and not a comma p. 90 Exercise 9.6. Change dented denoted p. 97 Change x x in equations (0.8), (0.0), (0.) and (0.2) 3

4 p. 207 In definition., change continuously differentiable over the closed and convex set continuously differentiable over the set. p. 207 Lemma.2. Change the problem from (G) (G) min h(x) s.t. x C, min f(x) s.t. x C, p. 22 In the last line, change the constraint x 2 + 2x x 3 3 = 2 x 2 + 2x x 2 3 p. 25 Change the first sentence of the proof of Theorem.3: As in the proof of the necessity of the KKT conditions (Theorem.5), By Theorem.4, p. 225 In the definition of I(x ), change f i (x ) = 0 g i (x ) = 0 p. 226 Change ( ) 0 2 xl(x, x 2, µ) ( ) 0 d 2 0 d 2 ( ) ( ) 0 d2 2 0 x L(x, x 2, µ) 0 d 2 p. 233 Line 4. Remove f from the chain of equalities/inequalities. p. 238 Line -6. Change λ, λ 2, λ m λ, λ 2,..., λ m p. 239 First line. Change there are there may be p. 239 In the premise of Theorem 2.2 change being finite-valued functions being functions p. 24 Just before Theorem 2.5, add the sentence Although the theorem holds for any convex set C, we will state and prove it only for convex sets with a nonempty interior p. 24 Theorem 2.5. Replace convex set and let with convex set with a nonempty interior and let The beginning of the proof of Theorem 2.5 should be: Although the theorem holds for any convex set C, we will prove it only for sets with a nonempty interior p. 242 In the statement of Theorem, 2.6, replace two nonempty convex sets such that with two convex sets with nonempty interiors such that 4

5 p. 242 First line of the proof of Theorem 2.6, after the parentheses write with a nonempty interior p. 242 First line. Replace Note that S, T are nonempty and convex and in addition with Note that S, T are convex with nonempty interiors and in addition p. 259 In the first displayed equation, change L(x, λ, λ 2,..., λ m ) L(x, y, y 2,..., y m, λ, λ 2,..., λ m ). p th line after (2.27). Remove (see also part (i) of Exercise.3) p. 267 At the displayed equation at the beginning of Section change the right-hand side of the inequality and equality constraints from 0. p. 272 In question 2.2, change Lat Let 5