Typos/Inconsistencies/Comments/Additions

1. Page 1, line -4: "... is not bounded below on ..."
2. Page 4, Regarding comments on Lagrange and Lagrange multipliers: in

   author    = " B.H. POURCIAU",
   title     = "Modern multiplier methods",
   journal   = "American Mathematical Monthly",
   year      = "1980",
   volume    = "87",
   number    = "6",
   pages     = "433-451"}
   
   

   it is stated that Lagrange multipliers are actually due to Euler and that Lagrange was Euler's student.
3. Page 18, Example 1.21: Both sets are NOT convex. The first set in R² needs the additional constraint x_2>=0. The second set in R³ should use sqrt(x_1^2+x_2^2) <= x_3 i.e. this guarantees that x_3 >=0.
4. Page 34, Theorem 2.5: The theorem holds for \bar{x} a **local** minimum of the function f, i.e. local is added and no assumption on convexity of f is needed. But with no convexity, the Remark following the theorem is not true, i.e. one cannot just use the unit vectors as directions. (An example was presented in class.)
5. Page 36, Theorem 29: We proved a stronger statement, called Pshnenichnyi's Condition, without the assumption that the function f is convex and over a general convex set. Then, with the assumption that f is convex, we proved sufficiency as well.
6. Page 43, Definition 2.20: THe definition of an Ideal Slater point in the notes is wrong. "An ideal Slater point is a Slater point x^* where g_j(x^*) l.t. 0, for all j in J_r" (A Slater point x^* must satisfy g_j linear for all j in J_s.)
7. We proved an additional Theorem: about strict separation of a point and a closed convex set by a hyperplane.
8. We use the hyperplane separation Theorem to prove the lemma: K is a c.c.c. iff K = polar of the polar of K. This proves the 'important, deep theorem' on Page 20.