We consider the problem of minimizing a polynomial function of degree three over the boundary of the sphere. If the objective is quadratic instead of cubic, this is the well-studied trust region subproblem, which is known to be tractable. In the cubic case, the problem turns out to be NP-hard. In this paper, we derive and evaluate different approaches for computing lower bounds for the cubic problem

We present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy

We present a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the Broyden–Fletcher–Goldfarb–Shanno method depends only on the product of the dimensionality

We consider a mini-batch stochastic Bregman proximal gradient method and a mini-batch stochastic Bregman proximal extragradient method for stochastic convex composite optimization problems. A simplified and unified convergence analysis framework is proposed to obtain almost sure convergence properties and expected convergence rates of the mini-batch stochastic Bregman proximal gradient method and its

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete

We consider an abstract minimization problem in reflexive Banach spaces together with a specific family of approximating sets, constructed by perturbing the cost functional and the set of constraints. For this problem, we state and prove various well-posedness results in the sense of Tykhonov, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, compactness

In this paper, we propose a new viscosity extragradient algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. We prove a strong convergence theorem under some appropriate conditions imposed on the parameters. Finally, we give some numerical experiments to illustrate the advantages of our proposed algorithms. The main results

We study a class of finite horizon optimal economic growth problems with nonlinear utility functions and linear production functions. By using a maximum principle in the optimal control theory and employing the special structure of the problems, we are able to explicitly describe the unique solution via input parameters. Economic interpretations of the obtained results and an open problem about the

A new numerical method is presented for bilevel programs with a nonconvex follower’s problem. The basic idea is to piecewise construct convex relaxations of the follower’s problems, replace the relaxed follower’s problems equivalently by their Karush–Kuhn–Tucker conditions and solve the resulting mathematical programs with equilibrium constraints. The convex relaxations and needed parameters are constructed

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific

We give regularity results for minimizers of two special cases of polyconvex functionals. Under some structural assumptions on the energy density, we prove that minimizers are either bounded, or have suitable integrability properties, by using the classical Stampacchia Lemma.

In this paper, we propose to study a wide class of combinatorial optimization problems called combinatorial optimization problems with labelling. First, we give a combinatorial method to deal with the labelling version of some classical combinatorial optimization problems including minimum vertex cover, maximum independent set, minimum dominating set and minimum set cover, and convert the labelling

This paper expands the existing sufficiency results for the strong minimality of an extremal of the Bolza problem. We cover the case where the strict Legendre Clebsh conditions are not strictly verified. An efficient, easy to use, algorithm to prove minimality is provided. It can be used on solutions with bang-bang control and does not require any local controllability property. The interval where

A new extragradient projection method, which does not require generalized monotonicity, is devised in this paper. In order to ensure its global convergence, we assume only that the Minty variational inequality has a solution. In particular, it applies to quasimonotone variational inequalities having a nontrivial solution.

We propose a nonmonotone trust region method for unconstrained optimization problems on Riemannian manifolds. Global convergence to the first-order stationary points is proved under some reasonable conditions. We also establish local R-linear, super-linear and quadratic convergence rates. Preliminary experiments show that the algorithm is efficient.

We develop an operator splitting approach to solve diffeomorphic matching problems for sequences of surfaces in three-dimensional space. The goal is to smoothly match, at very fast rate, finite sequences of observed 3D-snapshots extracted from movies recording the smooth dynamic deformations of “soft” surfaces. We have implemented our algorithms in a proprietary software installed at The Methodist

In this paper, we consider the convergence rate of the alternating direction method of multipliers for solving the nonconvex separable optimization problems. Based on the error bound condition, we prove that the sequence generated by the alternating direction method of multipliers converges locally to a critical point of the nonconvex optimization problem in a linear convergence rate, and the corresponding

In this paper, we study the complexity of the forward–backward splitting method with Beck–Teboulle’s line search for solving convex optimization problems, where the objective function can be split into the sum of a differentiable function and a nonsmooth function. We show that the method converges weakly to an optimal solution in Hilbert spaces, under mild standing assumptions without the global Lipschitz

In this paper, we introduce the concept of trade-off ratio function, which is closely related to the well-known Geoffrion’s proper efficiency for multi-objective optimization problems, and investigate its boundedness property. For linear multi-objective optimization problems, we show that the trade-off ratio function is bounded on the efficient solution set. For piecewise linear multi-objective optimization

This paper re-considers a recent analysis on the so-called Couplet–Heyman problem of least-thickness circular masonry arch structural form optimization and provides complementary and novel information and perspectives, specifically in terms of the optimization problem, and its implications in the general understanding of the Mechanics (statics) of masonry arches. First, typical underlying solutions

This article introduces the Generalized Minimum Branch Vertices problem. Given an undirected graph, where the set of vertices is partitioned into clusters, the Generalized Minimum Branch Vertices problem consists of finding a tree spanning exactly one vertex for each cluster and having the minimum number of branch vertices, namely vertices with degree greater than two. When each cluster is a singleton

In this paper, we propose two methods for solving unconstrained multiobjective optimization problems. First, we present a diagonal steepest descent method, in which, at each iteration, a common diagonal matrix is used to approximate the Hessian of every objective function. This method works directly with the objective functions, without using any kind of a priori chosen parameters. It is proved that

This paper focuses on a class of zero-norm composite optimization problems. For this class of nonconvex nonsmooth problems, we establish the Kurdyka–Łojasiewicz property of exponent being a half for its objective function under a suitable assumption and provide some examples to illustrate that such an assumption is not very restricted which, in particular, involve the zero-norm regularized or constrained

Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch

We consider a second-order differential equation governed by a quasiconvex function with a nonempty set of minimizers. Assuming that the gradient of this function is Lipschitz continuous, the existence of solutions to the gradient system is guaranteed. We study the asymptotic behavior of these solutions in continuous and discrete times. More precisely, we show that, if a solution is bounded, then it

We consider sensitivity of a semidefinite program under perturbations in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. When the coefficient matrices are perturbed, the optimal values can change discontinuously as explained in concrete examples. We show that the optimal value of such a semidefinite program changes continuously under conditions involving

In this study, a structural optimisation problem, addressed through a stochastic multi-objective approach, is formulated and solved. The problem deals with the optimal design of exoskeleton structures, conceived as vibration control systems under seismic loading. The exoskeleton structure is assumed to be coupled to an existing primary inner structure for seismic retrofit: the aim is to limit the dynamic

We suggest a new approach to creation of general market equilibrium models involving economic agents with local and partial knowledge about the system and under different restrictions. The market equilibrium problem is then formulated as a quasi-variational inequality that enables us to establish existence results for the model in different settings. We also describe dynamic processes, which fall into

The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity

In this paper, a particular discrete-time nonlinear and time-invariant system represented as a vector difference equation is analyzed for its stability properties. The motivation for analyzing this particular system is that it models gated recurrent unit neural networks commonly used and well known in machine learning applications. From the technical perspective, the analyses exploit the systems similarities

Backward error analysis reveals the numerical stability of algorithms and provides elaborate stopping criteria for iterative methods. Compared with numerical linear algebra problems, the backward error analysis for optimization problems is more rarely conducted in the literature. This paper is devoted to the backward error analysis for several generalizations of tensor complementarity problems. We

Having a convex cone K in an infinite-dimensional real linear space X, Adán and Novo stated (in J Optim Theory Appl 121:515–540, 2004) that the relative algebraic interior of K is nonempty if and only if the relative algebraic interior of the positive dual cone of K is nonempty. In this paper, we show that the direct implication is not true even if K is closed with respect to the finest locally convex

This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease in the gradient norm. This work is based on the performance estimation problem approach. The worst-case gradient bound of the resulting method is optimal up to a constant for large-dimensional smooth convex minimization problems

In this paper, we consider a class of nonlinear systems driven by measures generalizing the class of impulsive systems. We use measures as control and prove existence of optimal controls (measures) and present necessary conditions of optimality. We apply the general results to derive necessary conditions of optimality for purely impulse-driven systems. These results are then applied to optimal control

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form \(n=6k\) for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that

Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming problems. For each problem, we study the intersection between a sphere and a Voronoi-like polyhedron. That way, we get information about a possible overlap between

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’

In the present paper, it is provided a representation result for the weak solutions of a class of evolutionary Hamilton–Jacobi–Bellman equations on infinite horizon, with Hamiltonians measurable in time and fiber convex. Such Hamiltonians are associated with a—faithful—representation, namely involving two functions measurable in time and locally Lipschitz in the state and control. Our results concern

This paper focuses on designing a unified approach for computing the projection onto the intersection of a one-norm ball or sphere and a two-norm ball or sphere. We show that the solutions of these problems can all be determined by the root of the same piecewise quadratic function. We make use of the special structure of the auxiliary function and propose a novel bisection algorithm with finite termination

We study the problem of extending continuous quasiconvex real-valued functions from a subspace of a real normed linear space. Our results are essentially finite-dimensional and are based on a technical lemma which permits to “extend” a nested family of open convex subsets of a given subspace to a nested family of open convex sets in the whole space, in such a way that certain topological conditions

In this paper, we introduce two new types of conditional random set taking values in a Banach space: the conditional interior and the conditional closure. The conditional interior is a version of the conditional core, as introduced by A. Truffert and recently developed by Lépinette and Molchanov, and may be seen as a measurable version of the topological interior. The conditional closure is a generalization

This paper deals with the existence of solutions to equilibrium and quasi-equilibrium problems without any convexity assumption. Coverage includes some equivalences to the Ekeland variational principle for bifunctions and basic facts about transfer lower continuity. An application is given to systems of quasi-equilibrium problems.

We study the problem of bringing a linear chain of masses connected by springs to an equilibrium in finite time by means of a control force applied to the first mass. We describe explicitly the desired feedback control and establish its local equivalence to the minimum-time one. We prove the robustness of the control with respect to unknown disturbances and compute the time of transfer as well as its

Structural monitoring plays a central role in civil engineering; in particular, optimal sensor positioning is essential for correct monitoring both in terms of usable data and for optimizing the cost of the setup sensors. In this context, we focus our attention on the identification of the dynamic response of beam-like structures with uncertain damages. In particular, the non-localized damage is described

In this paper, we present a Frank–Wolfe-type theorem for nonconvex cubic programming problems. This result is a direct extension of the previous ones by Andronov et al. (Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika 4:194–197, 1982) and Flores-Bazán et al. (Math. Program. 145:263–290, 2014). Under suitable conditions, we characterize the compactness of the solution set of cubic programming

We consider an optimal dividend and capital structure problem for a firm, which holds a certain amount of debt to which is associated a financial ratio covenant between the firm and its creditors. We study optimal policies under a bankruptcy framework, using a mixed reduced and structural approach in modeling default and liquidation times. Once in default, the firm is given a grace period during which

This work concerns with discrete-time Markov decision processes on a denumerable state space. Assuming that the decision maker is risk-averse with constant risk-sensitivity coefficient, the performance of a control policy is measured by an average criterion associated with a non-negative and bounded cost function. Under conditions ensuring that the optimal average cost is constant, but not necessarily

The development of additive manufacturing techniques for composite structures brought the emergence of a new class of composite materials: the variable angle-tow composites. Additive manufacturing of reinforced polymers allows the tow to be placed along a curvilinear path in each lamina. Accordingly, optimised solutions with enhanced properties can be manufactured. In this work, the multi-scale two-level

In this paper, we establish strong duality between affinely adjustable two-stage robust linear programs and their dual semidefinite programs under a general uncertainty set, that covers most of the commonly used uncertainty sets of robust optimization. This is achieved by first deriving a new version of Farkas’ lemma for a parametric linear inequality system with affinely adjustable variables. Our

We propose a new method to approximate curves that interpolate a given set of time-labeled data on Riemannian symmetric spaces. First, we present our new formulation on the Euclidean space as a result of minimizing the mean square acceleration. This motivates its generalization on some Riemannian symmetric manifolds. Indeed, we generalize the proposed solution to the the special orthogonal group, the

We review and discuss results obtained through an application of tools of nonlinear optimal control to biomedical problems. We discuss various aspects of the modeling of the dynamics (such as growth and interaction terms), modeling of treatment (including pharmacometrics of the drugs), and give special attention to the choice of the objective functional to be minimized. Indeed, many properties of optimal

In this paper, a nonzero-sum stochastic differential reinsurance game is studied. A model including controls for the market share (advertising), investment, and reinsurance policies is considered. A jump–diffusion process is used to represent insurance claims. Necessary conditions that would lead to the Nash equilibrium can be found in the duopoly game we consider. Cases with and without controls on

The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone, which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is conducted. With this tool, we were able

The present paper is devoted to building degree theory for a generalized mixed quasi-variational inequality in finite dimensional spaces. Then, by employing the obtained results, we prove the existence and stability of solutions to the considered generalized mixed quasi-variational inequality.

In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis.

The paper deals with a Bolza optimal control problem for a dynamical system, whose motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this problem, the Cauchy problem for the Hamilton–Jacobi–Bellman equation with coinvariant derivatives is considered. Minimax and viscosity solutions of the Cauchy problem

We investigate the structure of the local and global minimizers of the Huber loss function regularized with a sparsity inducing L0 norm term. We characterize local minimizers and establish conditions that are necessary and sufficient for a local minimizer to be strict. A necessary condition is established for global minimizers, as well as non-emptiness of the set of global minimizers. The sparsity

Motivated by the recent works on proximity operators and isotone projection cones, in this paper, we discuss the isotonicity of the proximity operator in quasi-lattices, endowed with general cones. First, we show that Hilbert spaces, endowed with general cones, are quasi-lattices, in which the isotonicity of the proximity operator with respect to one order and two mutually dual orders is then, respectively