Related papers: Enhanced Fritz John Stationarity, New Constraint Q…
This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality…
Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed…
Optimization theory in Banach spaces suffers from the lack of available constraint qualifications. Despite the fact that there exist only a very few constraint qualifications, they are, in addition, often violated even in simple…
Motivated by a growing list of nontraditional statistical estimation problems of the piecewise kind, this paper provides a survey of known results supplemented with new results for the class of piecewise linear-quadratic programs. These are…
In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at the referee point. Some second-order constraint…
The disjunctive system is a system involving a disjunctive set which is the union of finitely many polyhedral convex sets. In this paper, we introduce a notion of the relaxed constant positive linear dependence constraint qualification…
We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the…
Let $f,g_1,\dots,g_m$ be polynomials of degree at most $d$ with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Assume that $f$ is non-negative on a basic semi-algebraic set $S$ defined by polynomial inequalities $g_j(x)\ge…
This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: minimize f(x):=h(c(x)), where h is an infinite-valued proper convex function and c is C^2-smooth. We focus on the case…
We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse…
As a starting point of our research, we show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to…
Sparsity constrained minimization captures a wide spectrum of applications in both machine learning and signal processing. This class of problems is difficult to solve since it is NP-hard and existing solutions are primarily based on…
This paper provides second-order optimality conditions for optimization problems with generalized equation constraints (GEPs), a framework that encompasses several important and challenging models in mathematical programming, including…
The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive…
Consider a polynomial optimization problem. Adding polynomial equations generated by the Fritz John conditions to the constraint set does not change the optimal value. As proved in [arXiv:2205.04254 (2022)], the objective polynomial has…
Approximate stationarity conditions provide necessary optimality conditions without requiring additional assumptions by demanding that a perturbed stationarity system possesses solutions as the involved perturbations tend to zero. Together…
We show that the Mordukhovich-stationarity system associated with a mathematical program with complementarity constraints (MPCC) can be equivalently written as a system of discontinuous equations which can be tackled with a semismooth…
In the past years, augmented Lagrangian methods have been successfully applied to several classes of non-convex optimization problems, inspiring new developments in both theory and practice. In this paper we bring most of these recent…
Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution,…
This expository paper contains a concise introduction to some significant works concerning the Karush-Kuhn-Tucker condition, a necessary condition for a solution in local optimality in problems with equality and inequality constraints. The…