MOAOCFGD

Implements MOAOCFGD, an adaptive-order Caputo fractional gradient descent method for multi-objective optimization.

For a vector objective \(f=(f_1,\dots,f_m)\) the method replaces each integer gradient with a regularized Caputo fractional gradient and then picks a single common descent direction by solving a quadratic subproblem that improves every objective at once. The fractional order \(\alpha\) and the smoothing parameter \(\beta\) are not fixed: the run is split into stages, and each stage \(s\) uses its own pair \((\alpha_s,\beta_s)\), so the memory weighting sharpens as the iterates approach a Pareto-critical point. The direction \(d^k\) is a convex combination of the fractional gradients (the multipliers come from the subproblem's KKT conditions), and the step size \(\eta_k\) is found by an Armijo-type backtracking line search that must decrease all objectives simultaneously.

\[ \begin{aligned} {}^{C}_{c}\nabla^{\alpha}_{x} f_{j,\alpha,\beta}(x) &= \mathrm{diag}\!\bigl({}^{C}_{c}\nabla^{\alpha}_{x} I(x)\bigr)^{-1}\Bigl[{}^{C}_{c}\nabla^{\alpha}_{x} f_{j}(x) + \beta\,\mathrm{diag}(|x-c|)\,{}^{C}_{c}\nabla^{1+\alpha}_{x} f_{j}(x)\Bigr] \\ (t^k, d^k) &= \arg\min_{(t,d)\,\in\,\mathbb{R}\times\mathbb{R}^n}\; t + \tfrac{1}{2}\,\|d\|^2 \quad \text{s.t. } {}^{C}_{c}\nabla^{\alpha}_{x} f_{j,\alpha,\beta}(x^k)^{\top} d - t \le 0,\;\; j=1,\dots,m \\ d^k &= -\sum_{j=1}^{m} \lambda_j^k\, {}^{C}_{c}\nabla^{\alpha}_{x} f_{j,\alpha,\beta}(x^k), \qquad \sum_{j=1}^{m}\lambda_j^k = 1,\;\; \lambda_j^k \ge 0 \\ \eta_k &= \max\bigl\{ r^{\,i} : i\in\mathbb{N}_0,\; f_j(x^k + r^{\,i} d^k) \le f_j(x^k) + \sigma\, r^{\,i}\, t^k \;\; \forall j \bigr\} \\ x^{k+1} &= x^k + \eta_k\, d^k \end{aligned} \]

where \({}^{C}_{c}\nabla^{\alpha}_{x}\) is the order-\(\alpha\) Caputo fractional gradient with lower terminal \(c\), \(f_{j,\alpha,\beta}\) is its \(\beta\)-regularized form (\(I(x)\) the integrand normalizer, \(|x-c|\) applied componentwise), \(\alpha\in(0,1]\) and \(\beta\in\bigl[\tfrac{1-\alpha}{2-\alpha},\infty\bigr)\) are the stage-wise order and smoothing parameters, \(d^k\) is the common descent direction with KKT multipliers \(\lambda_j^k\), \(t^k<0\) is the subproblem value at non-critical points, \(\sigma\in(0,1)\) is the Armijo constant, \(r\in(0,1)\) the backtracking factor, \(\eta_k\) the step size, and the loop stops when \(\|d^k\|<\epsilon\).

Reference: Barsha Shaw, Md Abu Talhamainuddin Ansary, "An Adaptive Order Caputo Fractional Gradient Descent Method for Multi-objective Optimization Problems", arXiv 2025. https://arxiv.org/abs/2507.07674

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