4. Unconstrained Optimization See Sydsaeter(2005, Chapter 3) and Chiang(1984, Chapter 9) If there is a neighborhood N(x*)of a with radius r such that z* is the maximum point of f on Mr(a*), then a* is a local maximum point of f For f: R-R, if Df( )=0, call i or (i, f()) a stationary point, and f(e) the stationary value. Given a stationary point i, there are three possible situations at i: local maximum, local minimum point, and a reflection point Example 2.10. Compare y=r2 with y=x3 at =0 Theorem 2.7.(Extreme-Value Theorem). For continuous f: R-R and compact AC Rn max f(ar) has at least one solution Theorem 2.8. Let ACRn (a)If a'is an interior solution of max then(FOC) Df(a*)=0 and(SONC)D2f(a*)<0 (b) If Df(a*)=0 and(SOSC) D f(*<0, 3 Mr(*)st. is the maximum point of f on Nr(a* (c) If f is concave on A, any point a E A satisfying Df(a*)=0 is a maximum point (d)If f is strictly quasi-concave, a local maximum over a convex set A is the unique global maximum Note: the FOC and SONC are not necessary for corner solutions; they are also not sufficient for local maximization, even for interior points Example 2.11. Find a maximum point for f(1, I2)=T2-4.2+3.1.T2-I24. Unconstrained Optimization See Sydsæter (2005, Chapter 3) and Chiang (1984, Chapter 9). If there is a neighborhood Nr(x∗) of x∗ with radius r such that x∗ is the maximum point of f on Nr(x∗), then x∗ is a local maximum point of f. For f : Rn → R, if Df(ˆx)=0, call xˆ or (ˆx, f(ˆx)) a stationary point, and f(ˆx) the stationary value. Given a stationary point x, ˆ there are three possible situations at xˆ : local maximum, local minimum point, and a reflection point. Example 2.10. Compare y = x2 with y = x3 at x = 0.  Theorem 2.7. (Extreme-Value Theorem). For continuous f : Rn → R and compact A ⊂ Rn, max x∈A f(x) has at least one solution.  Theorem 2.8. Let A ⊂ Rn. (a) If x∗ is an interior solution of max x∈A f(x) then (FOC) Df(x∗)=0 and (SONC) D2f(x∗) ≤ 0. (b) If Df(x∗)=0 and (SOSC) D2f(x∗) < 0, ∃ Nr(x∗) s.t. x∗ is the maximum point of f on Nr(x∗). (c) If f is concave on A, any point x∗ ∈ A satisfying Df(x∗)=0 is a maximum point. (d) If f is strictly quasi-concave, a local maximum over a convex set A is the unique global maximum.  Note: the FOC and SONC are not necessary for corner solutions; they are also not sufficient for local maximization, even for interior points. Example 2.11. Find a maximum point for f(x1, x2) = x2 − 4x2 1 + 3x1x2 − x2 2.  2—7