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

Abstract

Abstract We present and analyze a new generalized Frank–Wolfe method for the composite optimization problem $$(P): {\min }_{x\in {\mathbb {R}}^n} \; f(\mathsf {A} x) + h(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>P</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:msub><mml:mo>min</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:msub><mml:mspace/><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> , where f is a $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>θ</mml:mi></mml:math> -logarithmically-homogeneous self-concordant barrier, $$\mathsf {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is a linear operator and the function h has a bounded domain but is possibly non-smooth. We show that our generalized Frank–Wolfe method requires $$O((\delta _0 + \theta + R_h)\ln (\delta _0) + (\theta + R_h)^2/\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>θ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>θ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>/</mml:mo><mml:mi>ε</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> iterations to produce an $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> -approximate solution, where $$\delta _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>δ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> denotes the initial optimality gap and $$R_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> is the variation of h on its domain. This result establishes certain intrinsic connections between $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>θ</mml:mi></mml:math> -logarithmically homogeneous barriers and the Frank–Wolfe method. When specialized to the D -optimal design problem, we essentially recover the complexity obtained by Khachiyan (Math Oper Res 21 (2): 307–320, 1996) using the Frank–Wolfe method with exact line-search. We also study the (Fenchel) dual problem of ( P ), and we show that our new method is equivalent to an adaptive-step-size mirror descent method applied to the dual problem. This enables us to provide iteration complexity bounds for the mirror descent method despite the fact that the dual objective function is non-Lipschitz and has unbounded domain. In addition, we present computational experiments that point to the potential usefulness of our generalized Frank–Wolfe method on Poisson image de-blurring problems with TV regularization, and on simulated PET problem instances.

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