Compressed sensing involves solving a minimization problem with objective function $Ω(\boldsymbol{x}) = \|\boldsymbol{x}\|_1$ and linear constraints $\boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}$. Previous work has explored robustness to errors in $\boldsymbol{A}$ and $\boldsymbol{b}$ under special assumptions. Motivated by these results, we explore robustness to errors in $\boldsymbol{A}$ for a wider class of objective functions $Ω$ and for a more general setting, where the solution may not be unique. Similar results for errors in $\boldsymbol{b}$ are known and easier to prove. More precisely, for a seminorm $Ω(\boldsymbol{x})$ with a polyhedral unit ball, we prove that the set-valued map $S(\boldsymbol{A}) = \arg \min_{\boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}} Ω(\boldsymbol{x})$ is calm in $\boldsymbol{A}$, where calmness is a kind of local Lipschitz regularity.
The paper utilizes Hölder graphical derivatives for characterizing Hölder strong subregularity, isolated calmness and sharp minimum. As applications, we characterize Hölder isolated calmness in linear semi-infinite optimization and Hölder sharp minimizers of some penalty functions for constrained optimization.
This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for solutions of perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set map at the corres…
Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.
Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush--Kuhn--Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying…
The main purpose of this paper is to find conditions for Holder calmness of the solution mapping, viewed as a function of the boundary data, of a hemivariational inequality governed by the Navier-Stokes operator. To this end, a more abstract model is studied first: a class of parametric equilibrium problems defined by trifunctions. The presence of trifunctions allows the extension of the monotonicity notions and of the duality principle in the theory of equilibrium problems.
Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand-side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of …
This paper is devoted to studying the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed f…
In this paper, we provide a complete characterization on the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problem is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are sati…
In recent years there has been great interest in variational analysis of a class of nonsmooth functions called the minimal time function. In this paper we continue this line of research by providing new results on generalized differentiation of this class of functions, relaxing assumptions imposed on the functions and sets involved for the results. In particular, we focus on the singular subdifferential and the limiting subdifferential of this class of functions.
The partial calmness for the bilevel programming problem (BLPP) is an important condition which ensures that a local optimal solution of BLPP is a local optimal solution of a partially penalized problem where the lower level optimality constraint is moved to the objective function and hence a weaker constraint qualification can be applied. In this paper we propose a sufficient condition in the form of a partial error bound condition which guarantees the partial calmness condition. We analyse the partial calmness for the combined program based on the Bouligand (B-) and the Fritz John (FJ) stati…
The bilevel program is an optimization problem where the constraint involves solutions to a parametric optimization problem. It is well-known that the value function reformulation provides an equivalent single-level optimization problem but it results in a nonsmooth optimization problem which never satisfies the usual constraint qualification such as the Mangasarian-Fromovitz constraint qualification (MFCQ). In this paper we show that even the first order sufficient condition for metric subregularity (which is in general weaker than MFCQ) fails at each feasible point of the bilevel program. We…
On error bounds and noncritical Lagrange multipliers
Johann Radon Institute for Computational and Applied Mathematics（Johann Radon计算与应用数学研究所）；Dresden University of Technology, Institute of Numerical Mathematics（德累斯顿工业大学数值数学研究所）
The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Conseq…
In this paper, we exploit the so-called value function reformulation of the bilevel optimization problem to develop duality results for the problem. Our approach builds on Fenchel-Lagrange-type duality to establish suitable results for the bilevel optimization problem. First, we overview some standard duality results to show that they are not applicable to our problem. Secondly, via the concept of partial calmness, we establish weak and strong duality results. In particular, Lagrange, Fenchel-Lagrange, and Toland-Fenchel- Lagrange duality concepts are investigated for this type of problems und…
In this paper we introduce the notions of critical and noncritical multipliers for subdifferential variational systems extending to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush-Kuhn-Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax prob…
The main objective of this work is to study the existence of Lagrange multipliers for infinite dimensional problems under Gâteux differentiability assumptions on the data. Our investigation follows two main steps: the proof of the existence of Lagrange multipliers under a calmness assumption on the constraints and the study of sufficient conditions, which only use the Gâteaux derivative of the function defining the constraint, that ensure this assumption.
This paper concerns the study of a broad class of minimal time functions corresponding to control problems with constant convex dynamics and closed target sets in arbitrary Banach spaces. In contrast to other publications, we do not impose any nonempty interior and/or calmness assumptions on the initial data and deal with generally non-Lipschitzian minimal time functions. The major results present refined formulas for computing various subgradients of minimal time functions under minimal requirements in both cases of convex and nonconvex targets. Our technique is based on advanced tools of var…
The present paper contains some investigations about a uniform variant of the notion of metric hemiregularity, the latter being a less explored property obtained by weakening metric regularity. The introduction of such a quantitative stability property for set-valued mappings is motivated by applications to the penalization of constrained optimization problems, through the notion of problem calmness. As a main result, an implicit multifunction theorem for parameterized inclusion problems is established, which measures the uniform hemiregularity of the related solution mapping in terms of probl…
In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the…
We consider the optimistic bilevel optimization problem, known to have a wide range of applications in engineering, that we transform into a single-level optimization problem by means of the lower-level optimal value function reformulation. Subsequently, based on the partial calmness concept, we build an equation system, which is parameterized by the corresponding partial exact penalization parameter. We then design and analyze a Levenberg-Marquardt method to solve this parametric system of equations. Considering the fact that the selection of the partial exact penalization parameter is a crit…
The error bound property for a solution set defined by a set-valued mapping refers to an inequality that bounds the distance between vectors closed to a solution of the given set by a residual function. The error bound property is a Lipschitz-like/calmness property of the perturbed solution mapping, or equivalently the metric subregularity of the underlining set-valued mapping. It has been proved to be extremely useful in analyzing the convergence of many algorithms for solving optimization problems, as well as serving as a constraint qualification for optimality conditions. In this paper, we …
In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where the authors extend to Banach spaces the characterization of the strong regularity, we extend as well the characterizations for the metric subregularity and the strong subregularity given in [2] to Banach spaces. We also notice that at least one implication in these characterizations remains valid for the limiting subdifferential without assuming convexity o…
The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that t…
This paper is concerned with the rank constrained optimization problem whose feasible set is the intersection of the rank constraint set $\mathcal{R}=\!\big\{X\in\mathbb{X}\ |\ {\rm rank}(X)\le κ\big\}$ and a closed convex set $Ω$. We establish the local (global) Lipschitzian type error bounds for estimating the distance from any $X\in Ω$ ($X\in\mathbb{X}$) to the feasible set and the solution set, respectively, under the calmness of a multifunction associated to the feasible set at the origin, which is specially satisfied by three classes of common rank constrained optimization problems. As a…
Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various definitions for more than two decades, it has attracted much less attention than its older "siblings", the metric regularity and the strong metric regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable u…
