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<p>In this paper, we consider a class of difference-of-convex (DC) optimization problems, whose objective function is the difference of a relatively smooth convex function and a continuously convex function. We first propose a novel Bregman-Frank-Wolfe (BFW) algorithm for solving a DC optimization problem. In our algorithm, we mainly use the Bregman distance rather than the Euclidean distance in the selection of the step size. Moreover, we introduce an Adaptive BFW (ABFW) algorithm by adaptively selecting the step-size parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> that corresponds to the relatively smooth information of the objective function. The convergence analysis of the proposed algorithm can be established based on the triangle scaling property. We prove that all accumulation points of BFW and ABFW algorithms are stationary points. Then, we show that the convergence rates of the Frank-Wolfe (FW) gap of BFW and ABFW algorithms are both <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \mathcal {O} (k^{ \frac{1-\gamma }{\gamma }})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <mi>γ</mi> </mrow> <mi>γ</mi> </mfrac> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, some numerical experiments on non-convex quadratic inverse problems are conducted to demonstrate the effectiveness of our algorithms.</p>

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A Bregman-Frank-Wolfe Algorithm for DC Optimization Problems

  • 期刊论文

  • 发表日期：

    2026年6月25日

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  • Yao Zhang,
  • Bo Wen

摘要

In this paper, we consider a class of difference-of-convex (DC) optimization problems, whose objective function is the difference of a relatively smooth convex function and a continuously convex function. We first propose a novel Bregman-Frank-Wolfe (BFW) algorithm for solving a DC optimization problem. In our algorithm, we mainly use the Bregman distance rather than the Euclidean distance in the selection of the step size. Moreover, we introduce an Adaptive BFW (ABFW) algorithm by adaptively selecting the step-size parameter \(L_k\) L k that corresponds to the relatively smooth information of the objective function. The convergence analysis of the proposed algorithm can be established based on the triangle scaling property. We prove that all accumulation points of BFW and ABFW algorithms are stationary points. Then, we show that the convergence rates of the Frank-Wolfe (FW) gap of BFW and ABFW algorithms are both \( \mathcal {O} (k^{ \frac{1-\gamma }{\gamma }})\) O ( k 1 - γ γ ) . Finally, some numerical experiments on non-convex quadratic inverse problems are conducted to demonstrate the effectiveness of our algorithms.