Sufficient conditions for global weak Pareto solutions in multiobjective optimization

In this paper we derive new sufficient conditions for global weak Pareto solutions to set-valued optimization problems with general geometric constraints of the type $$\begin{aligned} \text{ maximize}\quad F(x) \quad \text{ subject} \text{ to}\quad x\in \Omega , \end{aligned}$$ where $F: X\rightrightarrows Z$ is a set-valued mapping between Banach spaces with a partial order on $Z$ . Our main results are established by using advanced tools of variational analysis and generalized differentiation; in particular, the extremal principle and full generalized differential calculus for the subdifferential/coderivative constructions involved. Various consequences and refined versions are also considered for special classes of problems in vector optimization including those with Lipschitzian data, with convex data, with finitely many objectives, and with no constraints.     

On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities

We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions u _λ for λ ∈ (?∞, λ*₀). The critical value λ*₀ >0 is found by a spectral analysis procedure according to Pokhozhaev’s fibering method. We show that the obtained solutions form a continuous branch (in the sense of level lines of the energy functional) with respect to the parameter λ. Moreover, we prove the existence of an interval \(( - \infty ,\tilde \lambda )\) , where \(\tilde \lambda > 0\) , on which this branch consists of solutions with exactly two nodal domains.     

Operational Calculus Approach to Nonlocal Cauchy Problems

Let Φ be a linear functional of the space ${\mathcal{C} =\mathcal{C}(\Delta)}$ of continuous functions on an interval Δ. The nonlocal boundary problem for an arbitrary linear differential equation $$ P\left(\frac{d}{d t}\right)y = F(t) $$ with constant coefficients and boundary value conditions of the form $$ \Phi\{\,y^{(k)}\} =\alpha_k,\,\,\,k = 0,\,1,\,2,\, \ldots,\,{\rm deg} P-1 $$ is said to be a nonlocal Cauchy boundary value problem. For solution of such problems an operational calculus of Mikusiński’s type, based on the convolution $$ (f*g)(t) = \Phi_\tau\, \left\{{\int\limits_\tau^t} f(t+\tau - \sigma)\,g(\sigma)\, d \sigma\, \right\}, $$ is developed. In the frames of this operational calculus the classical Heaviside algorithm is extended to nonlocal Cauchy problems. The obtaining of periodic, antiperiodic and mean-periodic solutions of linear ordinary differential equations with constant coefficients both in the non-resonance and in the resonance cases reduces to such problems. Here only the non-resonance case is considered. Extensions of the Duhamel principle are proposed.     

Thresholded covering algorithms for robust and max–min optimization

In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the \(k\) -robust model where the possible scenarios tomorrow are given by all demand-subsets of size \(k\) . In this paper, we give the following simple and intuitive template for \(k\) -robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for \(k\) -robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our \(k\) -robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, which are based on dual-rounding and primal–dual ideas; these properties might be of some independent interest. As a by-product of our techniques, we also get algorithms for max–min problems of the form: “given a covering problem instance, which \(k\) of the elements are costliest to cover?” For the problems mentioned above, we show that their \(k\) -max–min versions have performance guarantees similar to those for the \(k\) -robust problems.     

On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

Modal criteria for robust optimization of vibroacoustic systems

This work is an extension of the work done in previous papers (Besset and Jézéquel in Intern J Numer Method Eng 70(5):523–542, 2007; J Vib Acoust 130(1):011008, 2008; Intern J Numer Method Eng 73:1347–1373, 2008; J Vib Acoust 130(3):031009, 2008). It deals with modal criteria allowing to process optimization of structures including robustness. The system considered in the paper is a fluid-structure system. The aim of the paper is to use component mode synthesis methods to optimize the geometry of the structure and the robustness with respect to the design parameter variations of its vibroacoustic behaviour. Two criteria will be proposed. The first one is directly linked to the pressure level in the acoustic cavity. The second one is linked to the robustness of the methods. It comes from the polynomial chaos study of the considered system. A classical multiobjective optimization is then processed and Pareto graphics are proposed to represent the optimal solutions of the optimization problem.     

Parametric approaches to fractional programs

The fractional program P is defined by maxf(x)/g(x) subject tox∈X. A class of methods for solving P is based on the auxiliary problem Q(λ) with a parameter λ: maxf(x)?λg(x) subject tox∈X. Starting with two classical methods in this class, the Newton method and the binary search method, a number of variations are introduced and compared. Among the proposed methods. the modified binary search method is theoretically interesting because of its superlinear convergence and the capability to provide an explicit interval containing the optimum parameter value \(\bar \lambda \) . Computational behavior is tested by solving fractional knapsack problems and quadratic fractional programs. The interpolated binary search method seems to be most efficient, while other methods also behave surprisingly well.     

An approach to constrained global optimization based on exact penalty functions

In the field of global optimization many efforts have been devoted to solve unconstrained global optimization problems. The aim of this paper is to show that unconstrained global optimization methods can be used also for solving constrained optimization problems, by resorting to an exact penalty approach. In particular, we make use of a non-differentiable exact penalty function ${P_q(x;\varepsilon)}$ . We show that, under weak assumptions, there exists a threshold value ${\bar \varepsilon >0 }$ of the penalty parameter ${\varepsilon}$ such that, for any ${\varepsilon \in (0, \bar \varepsilon]}$ , any global minimizer of P _q is a global solution of the related constrained problem and conversely. On these bases, we describe an algorithm that, by combining an unconstrained global minimization technique for minimizing P _q for given values of the penalty parameter ${\varepsilon}$ and an automatic updating of ${\varepsilon}$ that occurs only a finite number of times, produces a sequence {x ^k } such that any limit point of the sequence is a global solution of the related constrained problem. In the algorithm any efficient unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. Some numerical experimentation confirms the effectiveness of the approach.     

Multiple Vortices in the Aharony–Bergman–Jafferis–Maldacena Model

Vortices in non-Abelian gauge field theory play important roles in confinement mechanism and are governed by systems of nonlinear elliptic equations of complicated structures. In this paper, we present a series of existence and uniqueness theorems for multiple vortex solutions of the BPS vortex equations, arising in the dual-layered Chern–Simons field theory developed by Aharony, Bergman, Jafferis, and Maldacena, over ${\mathbb{R}^2}$ and on a doubly periodic domain. In the full-plane setting, we show that the solution realizing a prescribed distribution of vortices exists and is unique. In the compact setting, we show that a solution realizing n prescribed vortices exists over a doubly periodic domain ${\Omega}$ if and only if the condition $$n < \frac{\lambda |\Omega|}{2 \pi}$$ holds, where ${\lambda >0 }$ is the Higgs coupling constant. In this case, if a solution exists, it must be unique. Our methods are based on calculus of variations.     

Factorization of a convolution-type operator

Three convolution-type equations are considered in the space of entire functions with topology ofd uniform convergence: $$\begin{gathered} M{_{\mu}{_1}} [f] \equiv \smallint _C f(z + t)d\mu _1 = 0, \hfill \\ M{_\mu{_1}} [f] \equiv \smallint _C f(z + t)d\mu _2 = 0, \hfill \\ M_\mu [f] \equiv \smallint _C f(z + t)d\mu = 0 \hfill \\ \end{gathered}$$ with respective characteristic functions L₁(λ), L₂(λ), L(λ)=L₁(λ)· L₂(λ), suppμ ?c, suppμ ₁ ?c, suppμ ₂ ?c. The necessary and sufficient conditions are found that every solutionf(z) of the equation M_μ[f[ can be written as a sumf ₁(z) +f ₂(z), wheref ₁(z) is the solution of the equation \(M{_\mu{_1}} [f] = 0\) ,f ₂(z) is the solution of the equation \(M{_\mu{_2}} [f] = 0\) .     

The Cauchy Problem for the Average Field Equation Describing a Model of a Magnetic Solid

We study a model of a magnetic solid treated as a system of particles with mechanical moment $\vec s,\vec s \in S^2$ , and magnetic moment $\vec \mu ,\vec \mu = \vec s$ , interacting with one another via the magnetic field, which determines variations in the mechanical moment of each particle. We study the system of integro-differential equations describing the evolution of the one-particle distribution function for this system of particles. We prove existence and uniqueness theorems for the generalized and the classical solution of the Cauchy problem for this system of equations. We also prove that the generalized solution continuously depends on the initial conditions.     

Solutions of one-dimensional boundary-value problems with a parameter in Sobolev spaces

We consider the most general boundary-value problems for systems of m ordinary first-order differential equations. For the solutions of these problems, we find sufficient conditions for the continuity in parameter in the Sobolev space $ {{\left( {W_p^n} \right)}^m} $ with $ n\in \mathbb{N} $ and 1 ≤ p < ∞.     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.     

Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

We consider the Willmore-type functional $$\mathcal{W}_{\gamma}(\Gamma):= \int\limits_{\Gamma} H^2 \; dA -\gamma \int\limits_{\Gamma} K \; dA,$$ where H and K denote mean and Gaussian curvature of a surface Γ, and ${\gamma \in [0,1]}$ is a real parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to ${\mathcal{W}_{\gamma}}$ and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.     

The space decomposition theory for a class of eigenvalue optimizations

In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. Here we apply the $\mathcal{U}$ -Lagrangian theory to a class of D.C. functions (the difference of two convex functions): the arbitrary eigenvalue function λ _i , with affine matrix-valued mappings, where λ _i is a D.C. function. We give the first-and second-order derivatives of ${\mathcal{U}}$ -Lagrangian in the space of decision variables R ^m when transversality condition holds. Moreover, an algorithm framework with quadratic convergence is presented. Finally, we present an application: low rank matrix optimization; meanwhile, list its $\mathcal{VU}$ decomposition results.     

Variational and optimal control problems with delayed argument

This paper deals with variational and optimal control problems with delayed argument and presents analogs of the classical necessary conditions for optimality for problems in (n + 1)-space. It is mainly concerned with the functional $$J(y) = \int_a^b {f[t,y(t\user2{--}\tau ),y(t),\dot y(t\user2{--}\tau ),\dot y(t)] dt} $$ There are no side conditions; τ is a positive real number; andy is a continuous piecewise smooth vector function havingn components. The fundamental lemma of the calculus of variations is used in deriving an analog of the Euler equations. The usual construction is utilized in obtaining analogs of the Weierstrass and Legendre conditions. Also found is a fourth necessary condition involving the least proper value associated with a boundary value problem related to the second variation. A sufficient condition is obtained by the use of a simple expansion method. The last station of the paper outlines an extension of a maximal principle obtained by Hestenes to control problems which involve delays in both the state variable and the control variable.     

On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation

The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the one-dimensional anharmonic oscillator with potential V (x, τ) depending on the slow time τ = ?t and with a small nonconservative term ?g( $ \dot x $ , x, τ), ? ? 1. This problem was discussed in numerous papers, and in some sense the present paper looks like a “methodological” one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form $ X\left( {\frac{{S\left( \tau \right) + \varepsilon \varphi \left( \tau \right)}} {\varepsilon },I\left( \tau \right),\tau } \right) $ , where the phase S, the “slow” parameter I, and the so-called phase shift ? are found from the system of “averaged” equations. The pragmatic result is that one can take into account the phase shift ? by considering it as a part of S and by simultaneously changing the initial data for the equation for I in an appropriate way.     

Solving inverse cone-constrained eigenvalue problems

We compare various algorithms for constructing a matrix of order $n$ whose Pareto spectrum contains a prescribed set $\Lambda =\{\lambda _1,\ldots , \lambda _p\}$ of reals. In order to avoid overdetermination one assumes that $p$ does not exceed $n^2.$ The inverse Pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations. We also address the issue of computing Lorentz spectra and solving inverse Lorentz eigenvalue problems.     

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P _λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.     

Semiclassics for Particles with Spin via a Wigner–Weyl-Type Calculus

We show how to relate the full quantum dynamics of a spin-½ particle on \({\mathbb{R}^d}\) to a classical Hamiltonian dynamics on the enlarged phase space \({\mathbb{R}^{2d} \times \mathbb{S}^{2}}\) up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for \({\mathbb{R}^d}\) (Folland, Harmonic Analysis on Phase Space, 1989; Lein, Weyl Quantization and Semiclassics, 2010) combined with the Stratonovich–Weyl calculus for SU(2) (Varilly and Gracia-Bondia, Ann Phys 190:107–148, 1989). For a specific class of Hamiltonians, including the Rabi- and Jaynes–Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern–Gerlach experiment.