This also applies to lower semi-continuous functions in the sense that a lower (upper) semi-continuous function f on a compact set B reaches its minimum (maximum). If f is continuous, then f is lower (upper) semi-continuous. ii) If f, g are lower semi-continuous and λ > 0, then λf and f +g are lower semi-continuous. iii). The function f : X →R is said to be successively lower semi-continuous at x0 ∈ X if and only if f(x0) ≤ lim.
Because f is lower semicontinuous at x0, for a given ε = 12(f(x0)−k) there exists a neighborhood W(x0) of x0 such that (by definition). Note: Letfi(x) be lower semicontinuous functions onX, i∈I. the pointwise supremum of a family of lower semicontinuous functions) is a lower semicontinuous function. The largest lower semicontinuous minorant of the function f : X → R is called the lower semicontinuous regularization of f and is denoted by f.
Remark: It exists as the pointwise supremum of those lower semi-continuous functions everywhere less than f (cf. Remark above). Since epi g is closed, then g is a lower semi-continuous function with g(x) ≤ f(x) and this implies g(x) ≤ f(x) because the definition of f as the largest lower semi-continuous minorant down and from here follows epig = epif ⊇ epif. X →Ris called weakly (sequentially) lower semi-continuous atx0 than for every row(xn)n∈N-converging weakly tox0, i.e. Comparing this with the definition of successively lower semi-continuous functions, one can notice that the only difference is that there was rather xn→x0 orxn * x0.
Note: Instead of "weak sequential lower semi-continuous" we just say. i) weakly (sequentially) closed if for every weakly convergent sequence xn * x, xn ∈D, x∈D, follows. ii) weakly sequentially compact if every sequence in D contains a weakly convergent subsequence whose weak limit belongs to D. Let X be a normed space, D a non-empty weak sequentially compact set, and f weak-below semicontinuous on D. Note: Obviously , f weak-below semicontinuous (weak (consecutively) lower semicontinuous) in a Banach space implies that f bearing is semi-continuous, but not the other way around.
We need to prove that the norm associated with a Banach space is a weak-lower semicontinuous function. Remark: Comparing Theorem 3.1 with the last one, it is not difficult to notice that it was the analogous claim for lower semicontinuous functions. Later we will see that this property (f∗∗ ≡f) is satisfied for every weak-lower semicontinuous and real function on X. iv) Let us consider an affine function on Rf(x) = mx+n.
Primal and Dual Optimization Problems
Between (P) and (P∗) there exists the relationship stated in the following statement. weak duality)For problems (P) and (P∗) we have. After the above construction of the dual problem, we obtain as dual in (−P∗) the following problem. Remark: As we noted before, for Banach spaces X and Y and f the right convex lower semicontinuous function, the weak-lower semicontinuous follows.
Of course, in this case the original function f in (P) must be convex (Φ(x,0) = f(x)) and lower semicontinuous.
Stability
The problem (P) is stable if and only if the following two conditions are simultaneously met. Then the following statements are equivalent to each other. iii) (P) is stable and has a solution. a) For XBanach space a convex, the lower semicontinuous functionalf is weak-lower semicontinuous (Satz 2.5 in [1], p. 91). Proving the stability directly by means of the definition is in many cases not so easy.
Let us further suppose that there exists an x0 ∈X such that the functionalΦ(x0,·) :Y →Ris ends and is continuous at0∈Y.
Optimality Conditions
In this chapter we will define the so-called Lagrangian functional in a general case and then show the connection between conjugate duality (Fenchel-Rockafellar duality) and the well-known Lagrangian duality. The assumption that Φ is convex and weak-lower semicontinuous on X ×Y gives that for all x∈X the function Φx :y→Φ(x, y) is convex and weak-lower semicontinuous and therefore (cf. Theorem 5.2) Φ∗∗x = Φx. Remark: Introducing the Lagrangian, we see that (P) and (P∗) are related to min-max problems and the weak duality relation.
L(x, y∗) known in game theory. Saddle Point Theorem) Let Φ :X×Y →R be convex and weak-bottom semicontinuous.
Case I
8 important special cases of dual optimization problems. ii) weak-low semi-continuous q with q6=±∞ implies weak-low semi-continuous Φ and Φ6=±∞. This means that Theorem 6.3 gives stability of (P) and. optimality conditions) The following conditions are equivalent to each other. We note that if i) f and g are convex, then q and thus Φ is convex. ii) f and g are convex and weak-low semicontinuous and f, g 6=±∞ then q and Φ are convex and weak-low semicontinuous and q, Φ6=±∞.
We can then use Theorem 8.1 (strong duality) and Theorem 8.2 (optimality conditions). strong duality) Assume that f and g are convex and −∞ <. optimality conditions).
Case II
Partial ordering is compatible with the structure of the vector space X in the sense. The partial ordering presented above also induces the following sets (i) C ={x∈X :x≥0} is the set of positive elements. This can be rewritten as this turns out to be correct. and χEy is the indicator function of the set Ey ={x∈X :x∈D, Bx≤y}.
Ey is closed and convex in X ∀y∈Y (and also weakly (sequentially) closed because we have that in the topological Hausdorff space X if E ⊂X is convex and closed then E is weakly (sequentially) closed). The function Φ :X×Y → R is convex and under semi-continuous iff is under semi-continuous (weak-below semi-continuous if f is weak-below semi-continuous) and also Φ6=±∞. Write Φ in the form Φ(x, y) = ˜f(x) +χE(x, y), where ˜f is convex and below semi-continuous (weak-below semi-continuous if f is weak-below semi-continuous is).
Moreover, χE(x, y) is convex and lower semicontinuous, since E is a closed convex set in X×Y, then {(x, y) :χE(x, y)≤k} is closed for all constants ∈R. We can also use other strong duality claims, such as Theorem 6.2 and Corollary 6.1, where we need the weak-lower semicontinuity of Φ that follows. ii) Furthermore, using Theorem 6.4 (optimality conditions), we can deduce the optimality conditions for our current problem (P) and (P∗). Therefore, we obtain from Theorem 7.2 the saddle theorem (replace y∗ with −y∗ and have y∗ ≥0 as usual in convex programming).
From the properties of the rates we have. i) Let X be a real reflexive Banach space and X∗ its dual. Because (y−x)TC(y−x)<0 the last term on the right-hand side above is less than or equal to 0, so off convexity follows, i.e. Remark: According to a remark in the lecture every sublinear function is convex and because the next exercise guarantees the sublinearity of the gauge of a convex set, it can be easily concluded that the gauge of a convex set is convex.
Let x1, x2 ∈ X be given and let the infimum be reached (in case the infimum is not reached, the considerations must be slightly modified: consider "infimum sequences". To derive the dual problem in (P) we calculate the conjugate of the perturbation function , Φ :X∗×Y1∗×..×Ym∗ →R, Introducing new variables pi := Aix−yi, i = 1, .., m, and considering x∗ = 0, we get after division of the terms in the above expression,
