--- title: Radial Epiderivative in Nonsmooth Optimization url: https://www.emergentmind.com/topics/radial-epiderivative type: topic --- # Radial Epiderivative in Nonsmooth Optimization Searching arXiv for papers on radial epiderivative and closely related notions. Search query: radial epiderivative nonsmooth optimization The radial epiderivative is a generalized derivative defined through the radial geometry of the epigraph. For a proper function \(f:\mathbb{R}^n\to\overline{\mathbb{R}}\) and a point \(\bar x\), it is specified by \[ \operatorname{epi} f^r(\bar x;\cdot)=R(\operatorname{epi} f;(\bar x,f(\bar x))), \] where the closed radial cone is \[ R(X;\bar{x})=\{w\in\mathbb{R}^n:\exists \lambda_n>0,\ x_n\in X,\ \lambda_n(x_n-\bar{x})\to w\} = \operatorname{cl}(\operatorname{cone}(X-\{\bar{x}\})). \] An equivalent analytic representation is \[ f^r(\bar{x};d)=\inf_{t>0}\liminf_{u\to d}\frac{f(\bar{x}+tu)-f(\bar{x})}{t}. \] In this form, the notion is explicitly not based on the conventional limit \(t\downarrow 0\); it is presented as a global, epigraph-based derivative concept for nonsmooth and nonconvex analysis, with applications to descent, global optimality, weak subgradients, and multiplier rules [2209.02118, 2509.01272]. ## 1. Geometric definition and basic construction The defining object is the radial cone of the epigraph. If \(f\) is proper, then \(f^r(\bar x;\cdot)\) is the function whose epigraph is exactly the radial cone to \(\operatorname{epi} f\) at \((\bar x,f(\bar x))\). The same literature also gives the equivalent formula \[ f^{r}(\bar{x};h) = \inf_{t > 0} \liminf_{u \rightarrow h} \frac{f(\bar{x}+ tu)- f(\bar{x})}{t}, \] which reformulates the epigraphic definition directly in terms of Newton quotients [2209.02118]. A function is called radially epidifferentiable at \(\bar x\) if \(f^r(\bar x;d)\) exists and is finite for every direction \(d\in\mathbb{R}^n\). For functions defined on a subset \(X\), a restricted-domain version \(f^{r_X}(\bar x;d)\) is introduced in order to accommodate non-open and discrete domains [2509.01272]. The infimum-over-\(t>0\) construction is the structural feature that distinguishes the radial epiderivative from conventional local directional derivatives. The formulation uses the entire radial geometry of the epigraph rather than only infinitesimal behavior near \(\bar x\). This is why the notion is repeatedly described as global or radial, and why it is deployed in settings where tangent-cone or neighborhood-based constructions are unavailable or too restrictive [2509.01272]. ## 2. Relations to directional derivatives, subderivatives, and subdifferentials The optimization papers place the radial epiderivative in a hierarchy of generalized derivatives. One recorded inequality chain is \[ f^r(\bar{x};x-\bar{x}) \le d f(\bar{x};x-\bar{x}) \le f'(\bar{x};x-\bar{x}) \le f^\circ(\bar{x};x-\bar{x}), \] where \(df\) denotes Rockafellar’s subderivative, \(f'\) the standard directional derivative, and \(f^\circ\) Clarke’s generalized derivative. For convex proper functions, the radial epiderivative agrees with the directional derivative: \[ f^r(\bar{x};x)=f'(\bar{x};x). \] The same sources therefore position \(f^r\) as a lower, more conservative object than the standard local generalized derivatives, while still reducing to classical formulas in special regular cases [2209.02118, 2509.01272]. A parallel Banach-space line of work does not introduce a named radial epiderivative, but it studies radial subderivatives and gives a dual characterization of radial directional growth from subdifferential data. For proper lower semicontinuous \(f:X\to]-\infty,+\infty]\), an abstract subdifferential \(\partial\) is considered under \[ \partial_{MR}f \subset \partial f \subset \partial_C f \] together with a Separation Principle, and the support function \[ f^{\partial}(x;u):=\sup\{(x^*,u):x^*\in\partial f(x)\} \] is introduced. The central duality statement is \[ \limsup_{x'\to_v x} f^{r}(x';\,u+a(x-x')) \;=\; \limsup_{x'\to_v x} f'(x';\,u+a(x-x')) \;=\; \limsup_{x'\to_v x} f^{\partial}(x';\,u+a(x-x')). \] Under radial accessibility, this yields \[ f^{r}(x;u)\le \inf_{a\ge 0}\ \limsup_{x'\to x} f^{\partial}(x';\,u+a(x-x')). \] The same paper also proves the existence of sequences \((x_n,x_n^*)\in\operatorname{gph}\partial f\) with \(x_n\to_v x\) and \(f(x_n)\to f(x)\) such that \[ f^{r}(x;u)\le \liminf_{n\to\infty}(x_n^*,\,u+a(x-x_n))\qquad \forall a\ge 0. \] This provides a subdifferential-based reconstruction of radial directional behavior and is explicitly described as epiderivative-like in content [1611.04045]. ## 3. Existence, regularity, and weak subgradients A principal existence criterion is lower Lipschitz behavior. In \(\mathbb{R}^n\), the papers state that \(f\) is radially epidifferentiable at \(\bar{x}\) if and only if \(f\) is lower Lipschitz at \(\bar{x}\), meaning that there exists \(L>0\) such that \[ f(x)-f(\bar{x})\ge -L\|x-\bar{x}\|\qquad \forall x. \] If \(f\) is proper and radially epidifferentiable at \(\bar x\), then \(f^r(\bar x;\cdot)\) is positively homogeneous and \(f\) is lower semicontinuous at \(\bar x\). The converse of lower semicontinuity is false; the function \[ f(x)=-\sqrt{|x|} \] is given as lower semicontinuous at \(0\) but not radially epidifferentiable there [2209.02118, 2509.01272]. The same literature develops weak subgradients. A pair \((v,c)\in\mathbb{R}^n\times\mathbb{R}_+\) is a weak subgradient of \(f\) at \(\bar x\) if \[ f(x)\ge f(\bar x)+\langle v,x-\bar x\rangle-c\|x-\bar x\| \quad\forall x. \] If \(f\) has a radial epiderivative at \(\bar x\) for every direction, then \[ \partial^w f^r(\bar x;0)=\partial^w f(\bar x). \] This identifies the weak subdifferential of the radial epiderivative at the origin with the weak subdifferential of the original function [2209.02118]. Constructive formulas are also given. For every \(\varepsilon>0\) and every unit direction \(h\), there exists \((v,c)\in \partial^w f(\bar x)\) such that \[ v=(c+f^r(\bar x;h)-\varepsilon)h, \] equivalently, \[ f^r(\bar x;h)=\langle v,h\rangle-c+\varepsilon. \] An \(\ell_1\)-norm analogue replaces \(h\) by \(\operatorname{Sgn}(h)\). These formulas are presented as explicit routes for computing weak subgradients from radial epiderivatives and vice versa, and they are part of the optimization motivation for the concept [2209.02118]. ## 4. Descent directions and global optimality The defining optimization consequence is the exact characterization of descent. In the unconstrained setting, if \(f\) is radially epidifferentiable at \(\bar x\), then \[ h \text{ is a descent direction for } f \text{ at } \bar x \iff f^r(\bar x;h)<0. \] For proper radially epidifferentiable functions, global optimality is characterized by the origin: \[ \bar x \text{ is a global minimizer } \iff f^r(\bar x;\cdot)\text{ is minimized at }0. \] The box-constrained line-search paper states the same criterion as: \(f\) attains its global minimum at \(\bar x\) if and only if \(f^r(\bar x;\cdot)\) attains its minimum at \(d=0\) [2209.02118, 2504.05090]. The constrained framework generalizes these ideas by redefining feasible directions radially. For a set \(S\subset\mathbb{R}^n\) and \(\bar x\in S\), \[ D(\bar{x})=\{d\in\mathbb{R}^n:\ d\neq 0,\ \exists \lambda>0,\ \bar{x}+\lambda d\in S\}, \] and \[ \operatorname{cl}(D(\bar{x}))=R(S;\bar{x}). \] Global descent directions are encoded by \[ F_1(\bar{x})=\{d:\ f^r(\bar{x};d)<0\}. \] The resulting minimum principle is \[ \bar{x}\text{ is a global minimum over }S \iff F_1(\bar{x})\cap D(\bar{x})=\emptyset, \] equivalently, \[ f^r(\bar{x};d)\ge 0 \quad \forall d\in D(\bar{x}). \] This replaces local tangent-cone stationarity by the absence of feasible radial directions producing an actual decrease at some positive step [2509.01272]. The same framework is explicitly designed to include discrete domains. One example uses the finite set \[ X=\{(0,4),(4,0),(4,4),(1,2),(2,1)\}. \] At \((2,1)\), the radial epiderivative along the only feasible direction \(d=(-1,1)\) is positive, showing global optimality; at \((1,2)\), it is negative, showing nonoptimality. The paper remarks that the directional derivative is not applicable in this setting [2509.01272]. ## 5. Multiplier rules, constraint qualifications, and algorithms For inequality constraints \(g_i(x)\le 0\), the radial framework introduces direction sets such as \[ G_1(\bar{x})=\{d:\ g_i^{r_X}(\bar{x};d)<0\ \forall i\}, \qquad \widetilde{G}_1(\bar{x})=\{d:\ g_i^{r_X}(\bar{x};d)\le 0\ \forall i\}, \] together with active-constraint variants. It also defines the radial gradient vector, for chosen directions \(d_1,\dots,d_k\), by \[ \nabla^r_d f(\bar{x}) = \big(f^r(\bar{x};d_1),\dots,f^r(\bar{x};d_k)\big)^\prime. \] This is explicitly not a gradient in the differential sense; it is a vector of radial epiderivative values along a chosen feasible basis [2509.01272]. Under the hypotheses that \(f,g_i\) are radially epidifferentiable, \(D(\bar x)\) is \(k\)-dimensional, and \[ D(\bar{x})\subseteq \widetilde{G}_1(\bar{x}), \] the radial Fritz John condition states that if \(\bar x\) is a global solution, then there exist multipliers \(v_0,v_1,\dots,v_m\) such that \[ v_0 \nabla^{r_X}_d f(\bar{x}) + \sum_{i=1}^m v_i \nabla^{r_X}_d g_i(\bar{x}) \ge 0, \] \[ v_0,v_i\ge 0,\qquad (v_0,v)\neq (0,0). \] If, additionally, for a linearly independent set of feasible directions \(d_1,\dots,d_k\), the corresponding radial gradient vectors of the constraints are linearly independent, then \(v_0>0\), which yields the KKT-type form. A converse sufficiency theorem is also given: if for every basis of feasible directions there exist multipliers satisfying \[ \nabla^{r_X}_d f(\bar{x})+\sum_{i=1}^m v_i\nabla^{r_X}_d g_i(\bar{x})\ge 0, \quad v_i\ge 0,\quad v\neq 0, \] then \(\bar{x}\) is a global minimum [2509.01272]. Algorithmic work uses the same derivative as a descent filter. In box-constrained optimization, with \[ X=\{x=(x_1,\ldots,x_n): a_i\le x_i\le b_i,\ i=1,\ldots,n\}, \] the approximate radial epiderivative procedure starts from \(\bar x\), a direction \(h\), \(t_0>0\), and \(\beta>0\), evaluates \[ y_0=\frac{f(\bar{x}+t_0h)-f(\bar{x})}{t_0}, \] then scans \[ t_{k+1}=t_0+(k+1)\beta,\qquad \tilde{y}_{k+1}=\frac{f(\bar{x}+t_{k+1}h)-f(\bar{x})}{t_{k+1}}. \] Whenever \(\tilde y_{k+1}N\). The same paper proves a general convergence theorem for infinite sequences generated by descent directions satisfying \(f^r(x_k;d_k)<0\): every cluster point \((\bar x,\bar d)\in X\times D\) satisfies \[ 0=f^r(\bar{x};\bar{d})\le f^r(\bar{x};d)\qquad \forall d\in D. \] For proper concave \(f\), with directions drawn from basis vectors and their negatives, every cluster point satisfies \(f^r(\bar x;\bar d)=0\), and \(\bar x\) is a global minimum over \(X\). The paper combines this descent test with cyclic coordinate search and particle swarm optimization, using the radial epiderivative to certify whether heuristic-generated directions are actually useful [2504.05090]. ## 6. Other meanings of the term and related radial derivative theories The literature represented here uses the expression in different ways. In nonsmooth optimization, the radial epiderivative is the epigraph-based object described above. In spherical-coordinate distribution theory, by contrast, the term is attached to radial differentiation of singular generalized functions rather than to optimization [1609.09771]. In the distributional setting of \(\mathbb{R}^m\), the radial derivative of the delta distribution is problematic because \(r=|\underline{x}|\) and \(\underline{\omega}=\underline{x}/|\underline{x}|\) are not smooth at the origin. The conclusion is that \[ \partial_r\delta(\underline{x}) \] cannot be an ordinary distribution; instead it belongs to a new class of continuous linear functionals called signumdistributions. The bridge to standard distribution theory is \[ \underline{\omega}\partial_r\delta(\underline{x})=\partial_{\underline{x}}\delta(\underline{x}), \] and the framework is designed precisely to make such radial objects calculable in spherical coordinates [1609.09771]. Closely related distributional work studies singular radial kernels such as \(1/r\) in three dimensions. The generalized second-order partial derivatives of \(1/r\) are defined as distributions because the classical derivatives have a nonintegrable \(1/r^3\)-type singularity at the origin. The standard formula \[ \bar\partial_i\bar\partial_j \frac1r = \operatorname{wlim}_{\epsilon\to 0} \left[ \frac{3x_ix_j-r^2\delta_{ij}}{r^5}\,O(r-\epsilon) -\frac{4\pi}{3}\,\delta_{ij}\,\delta(\mathbf r) \right] \] is explicitly identified as a spherical regularization formula, and non-spherical alternatives, including spheroidal and cylindrical regularizations, are derived. This is a generalized derivative theory for radial singularities, but it is not the optimization notion of epigraph-based radial epiderivative [1009.2480]. A further, again distinct, development appears in \(q\)-deformed Clifford and radial algebra theory. There the objective is an intrinsic \(q\)-deformation of the vector derivative on radial algebras. The authors explicitly reject the naive coordinatewise Jackson substitution \[ Q_x=\sum_j e_j D_{q,x_j} \] because it does not preserve radial subalgebras. Instead they construct a finite \(q\)-Cartan derivative \(\partial^Y_{x,q}\) from the radial scalar variables \[ r=x^2,\qquad s_i=\{x,y_i\},\qquad c_{ij}=\{y_i,y_j\}, \] and then pass to a direct-limit operator \(\partial_{x,q}:R(S)\to R(S)\). This is a theory of intrinsic \(q\)-radial vector derivatives on radial algebras, not of nonsmooth optimization [2605.00775]. This suggests that context is indispensable. In current optimization, “radial epiderivative” denotes a global epigraphic derivative used for descent, regularity, and optimality in nonsmooth nonconvex problems; in spherical distribution theory and radial algebra theory, the same phrase is tied instead to radial differentiation of singular objects or intrinsic radial operators. Source: https://www.emergentmind.com/topics/radial-epiderivative