Related papers: Optimality conditions based on the Fr\'echet secon…
This paper is concerned with the derivation of necessary conditions for the optimal shape of a design problem governed by a non-smooth PDE. The main particularity thereof is the lack of differentiability of the nonlinearity in the state…
In this paper, we examine higher order difference problems. Using the "squeezing" argument, we derive both Euler's condition and the transversality condition. In order to derive the two conditions, two needed assumptions are identified. A…
Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality…
In the last two decades, the sequential optimality conditions, which do not require constraint qualifications and allow improvement on the convergence assumptions of algorithms, had been considered in the literature. It includes the work by…
The standard assumption for proving linear convergence of first order methods for smooth convex optimization is the strong convexity of the objective function, an assumption which does not hold for many practical applications. In this…
In this paper, we present new second-order algorithms for composite convex optimization, called Contracting-domain Newton methods. These algorithms are affine-invariant and based on global second-order lower approximation for the smooth…
We present an optimization problem in infinite dimensions which satisfies the usual second-order sufficient condition but for which perturbed problems fail to possess solutions.
We show that the augmented primal-dual gradient algorithms can achieve global exponential convergence with partially strongly convex functions. In particular, the objective function only needs to be strongly convex in the subspace…
This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from countably many points). We…
In recent years, the success of deep learning has inspired many researchers to study the optimization of general smooth non-convex functions. However, recent works have established pessimistic worst-case complexities for this class…
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with…
In this paper, we introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is harmonized with the classical higher-order Fr\'echet…
We propose several new nonsmooth Newton methods for solving convex composite optimization problems with polyhedral regularizers, while avoiding the computation of complicated second-order information on these functions. Under the…
We study an optimal control problem with a quadratic cost functional for non-Newtonian fluids of differential type. More precisely, we consider the system governing the evolution of a second grade fluid filling a two-dimensional bounded…
Second-order optimization methods exhibit fast convergence to critical points, however, in nonconvex optimization, these methods often require restrictive step-sizes to ensure a monotonically decreasing objective function. In the presence…
We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local…
This paper is concerned with first- and second-order optimality conditions as well as the stability for non-smooth semilinear optimal control problems involving the $L^1$-norm of the control in the cost functional. In addition to the…
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the…
For a general nonlinear control system, we study the problem of small time local attainability of a target which is the closure of an open set. When the target is smooth and locally the sublevel set of a smooth function, we develop second…
In this article we propose a new approach to an analysis of DC optimization problems. This approach was largely inspired by codifferential calculus and the method of codifferential descent and is based on the use of a so-called affine…