Local Second-Order Weak Sharp Minima

• Local second-order weak sharp minima are defined by a quadratic growth inequality that extends classic sharp minimizer conditions to nonsmooth and constrained optimization.
• They link second-order generalized derivatives, graphical derivatives, and positive definiteness conditions to provide no-gap characterizations of local optimality.
• The theory unifies approaches in finite-dimensional nonsmooth settings, constrained programs, and nonconvex problems, offering practical insights into strong local minimality and metric subregularity.

Searching arXiv for the cited and closely related papers on local second-order weak sharp minima, sharp minima, and second-order optimality conditions. Local second-order weak sharp minima are local minimizers characterized by a quadratic growth inequality relative to the solution set. In the general set-constrained formulation studied for

$\min f(x)\quad \text{s.t.}\quad g(x)\in K,$

a feasible point $\bar x\in \Phi:=g^{-1}(K)$ is a local second-order weak sharp minimizer if there exist $\kappa>0$ and $\delta>0$ such that

$$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$

where $S:=\arg\min_{x\in\Phi}f(x)$ and $(x,S)=\inf_{u\in S}\|x-u\|$ (Ma et al., 16 Jul 2025). When $\bar x$ is locally isolated in $S$, this reduces to the classical local quadratic growth condition. In finite-dimensional nonsmooth optimization, the same local second-order phenomenon is formulated as strong local minimality: $$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$ so that local second-order weak sharpness and strong local minimizers become equivalent local viewpoints on quadratic growth (Chieu et al., 2019).

1. Local notion and its place among sharpness concepts

The literature distinguishes linear sharpness from quadratic sharpness. On a Riemannian manifold, local weak sharp minimality is defined by

$\bar x\in \Phi:=g^{-1}(K)$0

for all $\bar x\in \Phi:=g^{-1}(K)$1, where $\bar x\in \Phi:=g^{-1}(K)$2; equivalently, $\bar x\in \Phi:=g^{-1}(K)$3 is a local weak sharp minimizer iff it is a local minimizer of

$\bar x\in \Phi:=g^{-1}(K)$4

(Karkhaneei et al., 2018). In the singleton case, recent work on sharp local minimizers uses the local linear growth condition

$\bar x\in \Phi:=g^{-1}(K)$5

which is explicitly first-order in distance (Corella, 23 May 2026).

Local second-order weak sharp minima replace this linear estimate by quadratic growth. The relevant distance is the distance to the whole solution set $\bar x\in \Phi:=g^{-1}(K)$6, not merely to a single reference point, unless the minimizer is isolated (Ma et al., 16 Jul 2025). This places the notion between classical weak sharp minima and local second-order sufficient optimality conditions: it is stronger than mere local minimality, but it is still a local property and does not require global error bounds.

A recurrent terminological point is that some papers central to the subject do not use the phrase “weak sharp minima” as their main term. In particular, the finite-dimensional nonsmooth theory of (Chieu et al., 2019) is organized around strong local minimizers and quadratic growth, even though the property studied is the local second-order analogue commonly associated with weak sharpness. By contrast, (Corella, 23 May 2026) is explicitly about sharp minima, but the growth estimate there is linear, so it does not provide a literal second-order theory.

2. Graphical-derivative formulation of quadratic growth

A central second-order object in nonsmooth finite-dimensional analysis is the subgradient graphical derivative

$\bar x\in \Phi:=g^{-1}(K)$7

For smooth data this reduces to the Hessian: $\bar x\in \Phi:=g^{-1}(K)$8 Under subdifferential continuity, prox-regularity, and twice epi-differentiability at $\bar x\in \Phi:=g^{-1}(K)$9 for $\kappa>0$0, one has the identity

$\kappa>0$1

which links the graphical-derivative approach to second subderivative theory (Chieu et al., 2019).

The main sufficient condition for quadratic growth is a positive-definiteness requirement on $\kappa>0$2. If $\kappa>0$3 is proper l.s.c., $\kappa>0$4, $\kappa>0$5, and there exists $\kappa>0$6 such that

$\kappa>0$7

then $\kappa>0$8 is a strong local minimizer with any modulus in $\kappa>0$9. Moreover,

$\delta>0$0

with $\delta>0$1 (Chieu et al., 2019).

The same paper also proves that the pointwise positivity condition

$\delta>0$2

still implies strong local minimality. In this sense, positive definiteness of the subgradient graphical derivative is the nonsmooth analogue of Hessian positivity.

3. Strong metric subregularity and no-gap characterizations

A major structural theme is the equivalence between quadratic growth and strong metric subregularity of the subdifferential. For a set-valued mapping $\delta>0$3, strong metric subregularity at $\delta>0$4 for $\delta>0$5 is recalled through the criterion

$\delta>0$6

In the optimization setting, positivity of the subgradient graphical derivative implies

$\delta>0$7

hence strong metric subregularity of $\delta>0$8, which in turn yields quadratic growth (Chieu et al., 2019).

Under regularity assumptions, this implication becomes an exact equivalence. If $\delta>0$9 is proper l.s.c., $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$0, and $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$1 is subdifferentially continuous, prox-regular, and twice epi-differentiable at $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$2 for $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$3, then the following are equivalent:

1. $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$4 is a strong local minimizer.
2. $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$5 is a local minimizer and $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$6 is strongly metrically subregular at $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$7 for $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$8.
3. $$f(x)\ge f(\bar x)+\kappa\,[(x,S)]^2,\qquad \forall x\in \Phi\cap B_\delta(\bar x),$$9 is positive definite in the pointwise sense.
4. $S:=\arg\min_{x\in\Phi}f(x)$0 is positive definite in the uniform quadratic sense.

In that regime,

$S:=\arg\min_{x\in\Phi}f(x)$1

(Chieu et al., 2019). This is a no-gap second-order characterization of local quadratic growth.

The same paper extends the equivalence to variationally convex functions. In that class, strong local minimizer, strong metric subregularity of $S:=\arg\min_{x\in\Phi}f(x)$2, and the two positive-definiteness conditions on $S:=\arg\min_{x\in\Phi}f(x)$3 remain equivalent, and one obtains

$S:=\arg\min_{x\in\Phi}f(x)$4

(Chieu et al., 2019).

The literature also contains a cautionary point. An earlier “sufficient condition of the second kind,” requiring that for each unit $S:=\arg\min_{x\in\Phi}f(x)$5 there exists $S:=\arg\min_{x\in\Phi}f(x)$6 with $S:=\arg\min_{x\in\Phi}f(x)$7, is shown not to imply quadratic growth, even for a convex function. This establishes that the stronger quantified positivity conditions above are genuinely necessary for the no-gap theory (Chieu et al., 2019).

4. Constrained programs under MSCQ and classical second-order conditions

The graphical-derivative framework extends to constrained problems of the form

$S:=\arg\min_{x\in\Phi}f(x)$8

with $S:=\arg\min_{x\in\Phi}f(x)$9 twice continuously differentiable and $(x,S)=\inf_{u\in S}\|x-u\|$0 closed convex and $(x,S)=\inf_{u\in S}\|x-u\|$1-cone reducible. Writing

$(x,S)=\inf_{u\in S}\|x-u\|$2

the key qualification is MSCQ, the metric subregularity constraint qualification, which is weaker than Robinson’s CQ (Chieu et al., 2019).

Under MSCQ, the normal cone admits the exact formula

$(x,S)=\inf_{u\in S}\|x-u\|$3

near $(x,S)=\inf_{u\in S}\|x-u\|$4, and the subgradient graphical derivative becomes

$(x,S)=\inf_{u\in S}\|x-u\|$5

Using second-order calculus for $(x,S)=\inf_{u\in S}\|x-u\|$6-reducible sets, this expression can be written in terms of Lagrange multipliers and the critical cone (Chieu et al., 2019).

The resulting theorem gives a complete no-gap characterization. If $(x,S)=\inf_{u\in S}\|x-u\|$7 is stationary, MSCQ holds, and $(x,S)=\inf_{u\in S}\|x-u\|$8 is $(x,S)=\inf_{u\in S}\|x-u\|$9-cone reducible at $\bar x$0, then the following are equivalent:

1. $\bar x$1 is a strong local minimizer of the conic program.
2. $\bar x$2 is a local minimizer and $\bar x$3 is strongly metrically subregular at $\bar x$4 for $\bar x$5.
3. $\bar x$6 is positive definite in the pointwise sense.
4. There exists $\bar x$7 such that

$\bar x$8

1. The “sufficient condition of the second kind” holds.
2. The classical second-order sufficient condition holds:

$\bar x$9

1. There exists $S$0 such that

$S$1

Moreover,

$S$2

(Chieu et al., 2019).

This result extends classical second-order optimality theory beyond Robinson’s CQ. For nonlinear programming, under affine reduction, the additional curvature term $S$3 vanishes, and under CRCQ strong local minimizers coincide with tilt-stable minimizers (Chieu et al., 2019).

5. Nonconvex constraints, measure spaces, and other second-order frameworks

A broad nonconvex set-constrained theory is developed for

$S$4

where $S$5 and $S$6 are $S$7 and $S$8 is closed and possibly nonconvex. The decisive second-order objects are the outer second-order tangent set

$S$9

and the asymptotic second-order tangent cone

$$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$0

The paper emphasizes that $$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$1 may be empty and need not be convex, whereas $$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$2 is always a cone and

$$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$3

(Ma et al., 16 Jul 2025).

To accommodate nonconvex second-order geometry, the paper replaces the classical support function

$$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$4

by the lower generalized support function

$$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$5

with $$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$6, and equality with $$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$7 when the relevant set is convex. Under MSCQ and local second-order weak sharp minimality, explicit necessary conditions are derived in terms of $$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$8, for example

$$f(x)-f(\bar x)\ge \frac{\kappa}{2}\|x-\bar x\|^2\quad \text{for all }x\in \mathbb B_\delta(\bar x),$$9

whenever $\bar x\in \Phi:=g^{-1}(K)$00 (Ma et al., 16 Jul 2025). The same work also gives sufficient conditions that do not rely on convexity of $\bar x\in \Phi:=g^{-1}(K)$01, convexity or nonemptiness of $\bar x\in \Phi:=g^{-1}(K)$02, uniform second-order regularity of $\bar x\in \Phi:=g^{-1}(K)$03, or uniform approximation of critical cones.

A different no-gap second-order theory appears in sparse optimization over measures: $\bar x\in \Phi:=g^{-1}(K)$04 Here the main equivalence is between a structural condition on the optimal dual state $\bar x\in \Phi:=g^{-1}(K)$05, a no-gap second-order condition involving the weak* second subderivative of the lifted Radon norm, and a local quadratic growth estimate in the bounded Lipschitz norm: $\bar x\in \Phi:=g^{-1}(K)$06 The second-order positivity condition is

$\bar x\in \Phi:=g^{-1}(K)$07

and the theory shows that quadratic growth in $\bar x\in \Phi:=g^{-1}(K)$08 is equivalent to this no-gap condition under the stated regularity and structural assumptions (Wachsmuth et al., 2024).

Second-order local minimality also appears in free-discontinuity problems. For the planar Mumford–Shah functional,

$\bar x\in \Phi:=g^{-1}(K)$09

strict positivity of the second variation at a critical triple junction implies local minimality with respect to small $\bar x\in \Phi:=g^{-1}(K)$10-perturbations of the discontinuity set. The strict stability condition is

$\bar x\in \Phi:=g^{-1}(K)$11

and it yields a genuine local minimality inequality against nearby admissible competitors (Cristoferi, 2016). This is not formulated in weak-sharp-minima terminology, but it provides a clear second-variation instance of isolated local minimality with a quantitative second-order barrier.

6. Geometric interpretations, manifold analogues, and terminological boundaries

The geometric meaning of sharpness depends on the growth order. For sharp local minimizers in metric spaces, small Lipschitz perturbations with $\bar x\in \Phi:=g^{-1}(K)$12 do not move the minimizer locally: $\bar x\in \Phi:=g^{-1}(K)$13 Equivalent characterizations are given through metric slope lower bounds and a cone or cusp geometry of the graph, together with a tangent-plane rolling interpretation (Corella, 23 May 2026). The same paper explicitly notes that these are first-order sharpness conditions: the estimate is linear in distance, and no Hessian or quadratic-growth criterion appears.

On Riemannian manifolds, weak sharp minima are treated through Fréchet and limiting subdifferentials, contingent cones, and a local distance lemma in exponential coordinates. The paper proves the manifold distance-subdifferential formula

$\bar x\in \Phi:=g^{-1}(K)$14

and the directional estimate

$\bar x\in \Phi:=g^{-1}(K)$15

with equality in finite dimensions. If $\bar x\in \Phi:=g^{-1}(K)$16 is a local weak sharp minimizer with modulus $\bar x\in \Phi:=g^{-1}(K)$17, then

$\bar x\in \Phi:=g^{-1}(K)$18

and

$\bar x\in \Phi:=g^{-1}(K)$19

(Karkhaneei et al., 2018). The same work remarks that higher-order necessary conditions can be formulated by modifying the contingent directional derivative, but it does not prove an explicit second-order theorem.

These developments delimit the scope of “local second-order weak sharp minima.” The phrase is exact when the defining estimate is quadratic in the distance to the solution set, as in (Ma et al., 16 Jul 2025), or when the local property is expressed as quadratic growth or strong local minimality, as in (Chieu et al., 2019). It is only analogical when applied to linear sharpness results with pronounced geometry, such as cone or cusp conditions (Corella, 23 May 2026), or to manifold weak sharp minima where only first-order necessary conditions are established (Karkhaneei et al., 2018).

Taken together, the modern theory shows a coherent pattern. Quadratic growth can be encoded by second-order generalized derivatives, second-order tangent constructions, or second variations; it is tightly linked to metric regularity properties of the optimality mapping; and, under appropriate regularity, it admits no-gap characterizations that match classical Hessian-based criteria while remaining valid in nonsmooth, nonconvex, and infinite-dimensional settings (Chieu et al., 2019, Ma et al., 16 Jul 2025, Wachsmuth et al., 2024).