Problems with resource allocation constraints and optimization over the efficient set

The paper studies a nonlinear optimization problem under resource allocation constraints. Using quasi-gradient duality it is shown that the feasible set of the problem is a singleton (in the case of a single resource) or the set of Pareto efficient solutions of an associated vector maximization problem (in the case of $k>1$ resources). As a result, a nonlinear optimization problem under resource allocation constraints reduces to an optimization over the efficient set. The latter problem can further be converted into a quasiconvex maximization over a compact convex subset of $\mathbb{R }^k_+.$ Alternatively, it can be approached as a bilevel program and converted into a monotonic optimization problem in $\mathbb{R }^k_+.$ In either approach the converted problem falls into a common class of global optimization problems for which several practical solution methods exist when the number $k$ of resources is relatively small, as it often occurs.     

The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}

Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).     

Regularization and iterative methods for monotone inverse variational inequalities

We consider the monotone inverse variational inequality: find $x\in H$ such that $$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$ where $\Omega $ is a nonempty closed convex subset of a real Hilbert space $H$ and $f:H\rightarrow H$ is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.     

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results

In the first part of this work, we presented a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem (Kleniati and Adjiman in J Glob Optim, 2014). The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree, where two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. In the present paper, the theoretical properties of the proposed algorithm are investigated and finite \(\varepsilon \) -convergence to a global solution of the bilevel problem is proved. Thirty-four problems from the literature are tackled successfully.     

An integral equation and a representation for a Green's function

The final step in the mathematical solution of many problems in mathematical physics and engineering is the solution of a linear, two-point boundary-value problem such as $$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< x \hfill \\ (0) = 0, \dot u(x) = 0 \hfill \\ \end{gathered} $$ Such problems frequently arise in a variational context. In terms of the Green's functionG, the solution is $$u(t) = \int_0^x {G(t, y, x)g(y) dy} $$ It is shown that the Green's function may be represented in the form $$G(t,y,x) = m(t,y) - \int_y^x {q(s)m(t, s) m(y, s)} ds, 0< t< y< x$$ wherem satisfies the Fredholm integral equation $$m(t,x) = k(t,x) - \int_0^x k (t,y) q(y) m(y, x) dy, 0< t< x$$ and the kernelk is $$k(t, y) = min(t, y)$$      

Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type

In this paper we study the system $$\begin{aligned}&\min \biggl \{-\mathcal H u_i(x,t)-\psi _i(x,t),u_i(x,t)-\max _{j\ne i}(-c_{i,j}(x,t)+u_j(x,t))\biggr \}=0,\\&u_i(x,T)=g_i(x),\ i\in \{1,\ldots ,d\}, \end{aligned}$$ where \((x,t)\in \mathbb R ^{N}\times [0,T]\) . A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth, and structural assumptions on the data, i.e., on the operator \(\mathcal H \) and on continuous functions \(\psi _i\) , \(c_{i,j}\) , and \(g_i\) . A key aspect is that we make no sign assumption on the switching costs \(\{c_{i,j}\}\) and that \(c_{i,j}\) is allowed to depend on \(x\) as well as \(t\) . Using the comparison principle, the existence of a unique viscosity solution \((u_1,\ldots ,u_d)\) to the system is constructed as the limit of an increasing sequence of solutions to associated obstacle problems. Having settled the existence and uniqueness, we subsequently focus on regularity of \((u_1,\ldots ,u_d)\) beyond continuity. In this context, in particular, we assume that \(\mathcal H \) belongs to a class of second-order differential operators of Kolmogorov type of the form: $$\begin{aligned} \mathcal H =\sum _{i,j=1}^m a_{i,j}(x,t)\partial _{x_i x_j}+\sum _{i=1}^m a_i(x,t)\partial _{x_i} +\sum _{i,j=1}^N b_{i,j}x_i\partial _{x_j}+\partial _t, \end{aligned}$$ where \(1\le m\le N\) . The matrix \(\{a_{i,j}(x,t)\}_{i,j=1,\ldots ,m}\) is assumed to be symmetric and uniformly positive definite in \(\mathbb R ^m\) . In particular, uniform ellipticity is only assumed in the first \(m\) coordinate directions, and hence, \(\mathcal H \) may be degenerate.     

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

We study approximations by conforming methods of the solution to the variational inequality \({\langle \partial_t u,v-u\rangle + \psi(v) - \psi(u) \ge \langle f,v-u\rangle}\) , which arises in the context of inviscid incompressible Bingham type fluid flows and of the total variation flow problem. In the general context of a convex lower semi-continuous functional \({\psi}\) on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for nonsmooth data. Then, we introduce a general class of total variation functionals \({\psi}\) , for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms and prove their convergence for general total variation functionals. A comparison with an analytical solution concludes this study.     

On efficiency and mixed duality for a new class of nonconvex multiobjective variational control problems

In this paper, we extend the notions of \((\Phi ,\rho )\) -invexity and generalized \((\Phi ,\rho )\) -invexity to the continuous case and we use these concepts to establish sufficient optimality conditions for the considered class of nonconvex multiobjective variational control problems. Further, multiobjective variational control mixed dual problem is given for the considered multiobjective variational control problem and several mixed duality results are established under \((\Phi ,\rho )\) -invexity.     

Singular perturbation method for inhomogeneous nonlinear free boundary problems

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.     

A generalized min–max theorem for functionals of strongly indefinite sign

We consider non-linear Schrödinger equations of the following type: $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x) + V(x)u(x)-q(x)|u(x)|^\sigma u(x) = \lambda u(x), \quad x\in \mathbb{R }^N \\ u\in H^1(\mathbb{R }^N)\setminus \{0\}, \end{array} \right. \end{aligned}$$ where $N\ge 1$ and $\sigma >0$ . We will concentrate on the case where both $V$ and $q$ are periodic, and we will analyse what happens for different values of $\lambda $ inside a spectral gap $]\lambda ^-,\lambda ^+[$ . We derive both the existence of multiple orbits of solutions and the bifurcation of solutions when $\lambda \nearrow \lambda ^+$ . Thereby we use the corresponding energy function ${I_\lambda }$ and we derive a new variational characterization of multiple critical levels for such functionals: in this way we get multiple orbits of solutions. One main advantage of our new view on some specific critical values $c_0(\lambda )\le c_1(\lambda )\le \cdots \le c_n(\lambda )\le \cdots $ is a multiplicity result telling us something about the number of critical points with energies below $c_n(\lambda )$ , even if for example two of these values $c_i(\lambda )$ and $c_j(\lambda )$ ( $0\le i<j\le n$ ) coincide. Let us close this summary by mentioning another main advantage of our variational characterization of critical levels: we present our result in an abstract setting that is suitable for other problems and we give some hints about such problems (like the case corresponding to a Coulomb potential $V$ ) at the end of the present paper.