v2010.10.26 - Convex Optimization

240 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSconvex set C an intersection with a positive semidefinite cone, then thisproblem would be called a semidefinite program.There are two distinct ways to visualize this problem: one in the objectivefunction’s domain R 2 ,[ the]other including the ambient space of the objectiveR2function’s range as in . Both visualizations are illustrated in Figure 73.RVisualization in the function domain is easier because of lower dimension andbecauselevel sets (555) of any affine function are affine. (2.1.9)In this circumstance, the level sets are parallel hyperplanes with respectto R 2 . One solves optimization problem (552) graphically by finding thathyperplane intersecting feasible set C furthest right (in the direction ofnegative gradient −a (3.6)).When a differentiable convex objective function f is nonlinear, thenegative gradient −∇f is a viable search direction (replacing −a in (552)).(2.13.10.1, Figure 67) [152] Then the nonlinear objective function can bereplaced with a dynamic linear objective; linear as in (552).3.4.0.0.3 Example. Support function. [199,C.2.1-C.2.3.1]For arbitrary set Y ⊆ R n , its support function σ Y (a) : R n → R is definedσ Y (a) supa T z (553)z∈Ya positively homogeneous function of direction a whose range contains ±∞.[250, p.135] For each z ∈ Y , a T z is a linear function of vector a . Becauseσ Y (a) is a pointwise supremum of linear functions, it is convex in a(Figure 74). Application of the support function is illustrated in Figure 29afor one particular normal a . Given nonempty closed bounded convex sets Yand Z in R n and nonnegative scalars β and γ [371, p.234]σ βY+γZ (a) = βσ Y (a) + γσ Z (a) (554)3.4.0.0.4 Exercise. Level sets.Given a function f and constant κ , its level sets are definedL κ κf {z | f(z)=κ} (555)