Analysis of the Frank–Wolfe method for convex composite optimization involving a logarithmically-homogeneous barrier

Abstract We present and analyze a new generalized Frank–Wolfe method for the composite optimization problem $$(P): {\min }_{x\in {\mathbb {R}}^n} \; f(\mathsf {A} x) + h(x)$$ ( P ) : min x ∈ R n f A + h , where f is $$\theta $$ θ -logarithmically-homogeneous self-concordant barrier, $$\mathsf {A}$$ linear operator function h has bounded domain but possibly non-smooth. show that our requires $$O((\delta _0 \theta R_h)\ln (\delta _0) (\theta R_h)^2/\varepsilon )$$ O δ 0 ln 2 / ε iterations to produce an $$\varepsilon -approximate solution, $$\delta _0$$ denotes initial optimality gap $$R_h$$ variation of on its domain. This result establishes certain intrinsic connections between -logarithmically homogeneous barriers method. When specialized D -optimal design problem, we essentially recover complexity obtained by Khachiyan (Math Oper Res 21 (2): 307–320, 1996) using with exact line-search. also study (Fenchel) dual ( P ), equivalent adaptive-step-size mirror descent applied problem. enables us provide iteration bounds despite fact objective non-Lipschitz unbounded In addition, computational experiments point potential usefulness Poisson image de-blurring problems TV regularization, simulated PET instances.