v2007.09.17 - Convex Optimization

194 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSwhose objective of minimization is a real linear function. Were convex set Cpolyhedral (2.12), then this problem would be called a linear program.Were C a positive semidefinite cone, then this problem would be called asemidefinite program.There are two distinct ways to visualize this problem: one in theobjective function’s domain R 2 , the other [ including ] the ambient spaceR2of the objective function’s range as in . Both visualizations areRillustrated in Figure 55. Visualization in the function domain is easierbecause of lower dimension and because level sets of any affine function areaffine (2.1.9). In this circumstance, the level sets are parallel hyperplaneswith respect to R 2 . One solves optimization problem (457) graphically byfinding that hyperplane intersecting feasible set C furthest right (in thedirection of negative gradient −a (3.1.8)).When a differentiable convex objective function f is nonlinear, thenegative gradient −∇f is a viable search direction (replacing −a in (457)).(2.13.10.1, Figure 53) [104] Then the nonlinear objective function can bereplaced with a dynamic linear objective; linear as in (457).3.1.6.0.2 Example. Support function. [46,3.2]For arbitrary set Y ⊆ R n , its support function σ Y (a) : R n → R is definedσ Y (a) = ∆ supa T z (458)z∈Ywhose range contains ±∞ [182, p.135] [148,C.2.3.1]. For each z ∈ Y ,a T z is a linear function of vector a . Because σ Y (a) is the pointwisesupremum of linear functions, it is convex in a . (Figure 56) Applicationof the support function is illustrated in Figure 20(a) for one particularnormal a .3.1.7 epigraph, sublevel setIt is well established that a continuous real function is convex if andonly if its epigraph makes a convex set. [148] [230] [268] [280] [182]Thereby, piecewise-continuous convex functions are admitted. Epigraph isthe connection between convex sets and convex functions. Its generalizationto a vector-valued function f(X) : R p×k →R M is straightforward: [218]