Convex Optimization — Boyd & Vandenberghe 3. Convex …

Convex Optimization Boyd & Vandenberghe3. Convex functions basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions log-concave and log- Convex functions convexity with respect to generalized inequalities3 1 Definitionf:Rn Ris Convex ifdomfis a Convex set andf( x+ (1 )y) f(x) + (1 )f(y)for allx, y domf,0 1(x, f(x))(y, f(y)) fis concave if fis Convex fis strictly Convex ifdomfis Convex andf( x+ (1 )y)< f(x) + (1 )f(y)forx, y domf,x6=y,0< <1 Convex functions3 2 Examples on Rconvex: affine:ax+bonR, for anya, b R exponential:eax, for anya R powers:x onR++, for 1or 0 powers of absolute value:|x|ponR, forp 1 negative entropy:xlogxonR++concave: affine:ax+bonR, for anya, b R powers:x onR++, for0 1 logarithm:logxonR++ Convex functions3 3 Examples on Rnand Rm naffine functions are Convex and concave; all norms are convexexamples on Rn affine functionf(x) =aTx+b norms:kxkp= ( ni=1|xi|p)1/pforp 1;kxk = maxk|xk|examples on Rm n(m nmatrices) affine functionf(X) =tr(ATX) +b=m i=1n j=1 AijXij+b spectral (maximum singular value) normf(X) =kXk2= max(X) = ( max(XTX))1/2 Convex functions3 4 Restriction of a Convex function t

least-squares objective: f(x) = kAx−bk2 2 ... • nonnegative weighted sum • composition with affine function • pointwise maximum and supremum • composition • minimization • perspective Convex functions 3–13. Positive weighted sum & composition with affine function

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