Concave Term Research Articles

Singular non-autonomous [formula omitted]-equations with competing nonlinearities

We consider a parametric non-autonomous (p,q)-equation with a singular term and competing nonlinearities, a parametric concave term and a Carathéodory perturbation. We consider the cases where the perturbation is (p−1)-linear and where it is (p−1)-superlinear (but without the use of the Ambrosetti–Rabinowitz condition). We prove an existence and multiplicity result which is global in the parameter λ>0 (a bifurcation type result). Also, we show the existence of a smallest positive solution and show that it is strictly increasing as a function of the parameter. Finally, we examine the set of positive solutions as a function of the parameter (solution multifunction). First, we show that the solution set is compact in C01(Ω̄) and then we show that the solution multifunction is Vietoris continuous and also Hausdorff continuous as a multifunction of the parameter.

Anisotropic (p, q) Equation with Partially Concave Terms

We consider a Dirichlet problem driven by the anisotropic (p,q) Laplacian. In the reaction, we have a parametric partially concave term plus a “superlinear” perturbation (convex term) which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools, we show that for all small values of the parameter λ>0, the problem has at least two nontrivial smooth solutions.

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

Positive solutions for parametric equations with unbalanced growth and indefinite perturbation

We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one (“concave-convex” problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions.

An algorithm for stochastic convex-concave fractional programs with applications to production efficiency and equitable resource allocation

We propose an algorithm to solve convex and concave fractional programs and their stochastic counterparts in a common framework. Our approach is based on a novel reformulation that involves differences of square terms in the constraints, and subsequent employment of piecewise-linear approximations of the concave terms. Using the branch-and-bound (B&B) framework, our algorithm adaptively refines the piecewise-linear approximations and iteratively solves convex approximation problems. The convergence analysis provides a bound on the optimality gap as a function of approximation errors. Based on this bound, we prove that the proposed B&B algorithm terminates in a finite number of iterations and the worst-case bound to obtain an ϵ-optimal solution reciprocally depends on the square root of ϵ. Numerical experiments on Cobb–Douglas production efficiency and equitable resource allocation problems support that the algorithm efficiently finds a highly accurate solution while significantly outperforming the benchmark algorithms for all the small size problem instances solved. A modified branching strategy that takes the advantage of non-linearity in convex functions further improves the performance. Results are also discussed when solving a dual reformulation and using a cutting surface algorithm to solve distributionally robust counterpart of the Cobb–Douglas example models.

Convergence analysis of a positivity-preserving numerical scheme for the Cahn-Hilliard-Stokes system with Flory-Huggins energy potential

A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, is approximated in a semi-implicit manner. In the spatial discretization, the marker and cell difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis.

Optimal Design of Refinery Hydrogen Networks: Reduced Superstructure and Effective Algorithm

In refinery hydrogen networks, the cost related to hydrogen compressors is the second largest contributor to the total annualized cost (TAC). The optimal synthesis of hydrogen networks considering compressor arrangement is a computationally difficult optimization problem because of the highly nonlinear terms caused by hydrogen compression. In this study, a reduced superstructure for optimal design of hydrogen networks is first developed based on several heuristics. This reduced superstructure includes all possible connections among hydrogen sources, hydrogen sinks, and compressors. The maximum number of compressors is reduced dramatically, and the inlet and outlet pressures of all compressors can be preassigned, which notably reduces the bilinear terms and completely eliminates the linear fractional terms in the mathematical model. A mixed-integer nonlinear programming (MINLP) model is formulated, and a tailored algorithm that incorporates successive piecewise linear approximation of concave terms and linear relaxation of bilinear terms is proposed. The case studies show that the proposed model and algorithm can give better hydrogen network designs with only 2.9 and 4.9% of the computational time in the existing publication.

(p, q)-Equations with Negative Concave Terms

In this paper, we study a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction that has the combined effects of a negative concave term and of an asymmetric perturbation which is superlinear on the positive semiaxis and resonant in the negative one. We prove a multiplicity theorem for such problems obtaining three nontrivial solutions, all with sign information. Furthermore, under a local symmetry condition, we prove the existence of a whole sequence of sign-changing solutions converging to zero in C^1_0(overline{Omega }).

On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth

Abstract In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -\Delta u+V\left(x)u=f\left(x,u)+\lambda a\left(x)| u{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where λ > 0 \lambda \gt 0 , q ∈ ( 1 , 2 ) q\in \left(1,2) , a ∈ L 2 / ( 2 − q ) ( R 2 ) a\in {L}^{2\text{/}\left(2-q)}\left({{\mathbb{R}}}^{2}) , V ( x ) V\left(x) , and f ( x , t ) f\left(x,t) are 1-periodic with respect to x x , and f ( x , t ) f\left(x,t) has critical exponential growth at t = ∞ t=\infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ > 0 \lambda \gt 0 ) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.

Active-set algorithm-based statistical inference for shape-restricted generalized additive Cox regression models

Recently the shape-restricted inference has gained popularity in statistical and econometric literature to relax the linear or quadratic covariate effect in regression analyses. The typical shape-restricted covariate effect includes monotone increasing, decreasing, convexity or concavity. In this paper, we introduce the shape-restricted inference to the celebrated Cox regression model (SR-Cox), in which the covariate response is modelled as shape-restricted additive functions. The SR-Cox regression approximates the shape-restricted functions using a spline basis expansion with data-driven choice of knots. The underlying minimization of negative log-likelihood function is formulated as a convex optimization problem, which is solved with an active-set optimization algorithm. The highlight of this algorithm is that it eliminates the superfluous knots automatically. When covariate effects include combinations of convex or concave terms with unknown forms and linear terms, the most interesting finding is that SR-Cox produces accurate linear covariate effect estimates which are comparable to the maximum partial likelihood estimates if indeed the forms are known. We conclude that concave or convex SR-Cox models could significantly improve nonlinear covariate response recovery and model goodness of fit.

Double phase equations with an indefinite concave term

We consider a Dirichlet problem having a double phase differential operator with unbalanced growth and reaction involving the combined effects of a concave (sublinear) and of a convex (superlinear) terms. We allow the coefficient \(\mathcal E\in L^\infty(\Omega)\) of the concave term to be sign changing. We show that when \(\|\mathcal E\|_\infty \) is small the problem has at least two bounded positive solutions.

Solving the reliability-oriented generalized assignment problem by Lagrangian relaxation and Alternating Direction Method of Multipliers

The well-known generalized assignment problem has many real-world applications. The assignment costs between agents and tasks affected by several factors could be unstable and uncertain. In this paper, we assume that the means and variances of assignment costs are known in advance. The idea is that the decision-maker aims to minimize the total assignment costs not only on average, but also to keep their variability as small as possible. Then, a reliability-oriented model with a nonlinear and concave objective function is formulated, and two decomposition-based methods are systematically developed for this challenging problem. The task allocation constraints are dualized and the Lagrangian relaxed problem is broken into many reliability-oriented knapsack subproblems. By the Lagrangian substitution technique, each subproblem is further decomposed into a standard knapsack problem and a simple univariate concave minimization problem. A lower bound is constructed and multipliers are optimized in dual problems by the subgradient method. Feasible solutions can be generated from the results of these reliability-oriented subproblems by Lagrangian heuristic. To further improve the convergence and solution quality, an Alternating Direction Method of Multipliers (ADMM) based decomposition approach is proposed. The augmented Lagrangian relaxed problem is split into a sequence of subproblems by the block coordinate descent method. To cope with the quadratic terms and the concave terms in these subproblems, linearization and Lagrangian substitution techniques are applied. Feasible solutions are produced from these subproblems, and solution quality can be evaluated by the lower bound provided by the Lagrangian relaxed problem. Numerical experiments are conducted on the test cases transformed from the standard benchmark instances, and the ADMM-based method has superior convergence and solution quality.

Schrödinger equations in [formula omitted] with critical exponential growth and concave nonlinearities

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.

Multiple positive solutions of Kirchhoff-type equations with concave terms

Multiple positive solutions of Kirchhoff-type equations with concave terms

Energy-stable backward differentiation formula type fourier collocation spectral schemes for the Cahn-Hilliard equation

We present a variant of second order accurate in time backward differentiation formula schemes for the Cahn-Hilliard equation, with a Fourier collocation spectral approximation in space. A three-point stencil is applied in the temporal discretization, and the concave term diffusion term is treated explicitly. An addition-al Douglas-Dupont regularization term is introduced, which ensures the energy stability with a mild requirement. Various numerical simulations including the verification of accuracy, coarsening process and energy decay rate are presented to demonstrate the efficiency and the robustness of proposed schemes.

On a Class of Quasilinear Equations Involving Critical Exponential Growth and Concave Terms in $${\mathbb {R}}^N$$

On a Class of Quasilinear Equations Involving Critical Exponential Growth and Concave Terms in $${\mathbb {R}}^N$$

Nonlinear concave-convex problems with indefinite weight

We consider a parametric nonlinear Robin problem driven by the p-Laplacian and with a reaction having the competing effects of two terms. One is a parametric -sublinear term (concave nonlinearity) and the other is a -superlinear term (convex nonlinearity). We assume that the weight of the concave term is indefinite (that is, sign-changing). Using the Nehari method, we show that for all small values of the parameter , the problem has at least two positive solutions and also we provide information about their regularity.

A new Lagrangian-Benders approach for a concave cost supply chain network design problem

This article presents an extended facility location model for multiproduct supply chain network design, that accounts for concave capacity, transportation, and inventory costs (induced by economies of scale, quantity discounts and risk pooling, respectively). The problem is formulated as a mixed-integer nonlinear program (MINLP) with linear constraints and a large number of separable concave terms in the objective function. We propose a solution approach that combines stabilized Lagrangian relaxation with a novel Benders decomposition. Lagrangian relaxation is used first to decompose the problem by potential warehouse location into low-rank concave minimization subproblems. Benders decomposition is then applied to solve the subproblems by shifting the concave terms to a low-dimensional master problem that is solved effectively through implicit enumeration. Finally, a feasible solution for the original problem is constructed from the partial subproblems solutions by solving a restricted set covering problem. The proposed approach is tested on several instances from the literature and compared against a state-of-the-art solver. Furthermore, a realistic case study is used to demonstrate the impact of concavities in the cost components with extensive sensitivity analyses.

Singular (p,q)-equations with competing perturbations

We consider a nonlinear Dirichlet problem driven by the -Laplacian and with a reaction involving the combined effects of a singular term, of a concave term and of a parametric convex term. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies. Also, we show that for every admissible parameter , the problem has a minimal positive solution and we establish the monotonicity and continuity properties of the map .

On the set of positive solutions for resonant Robin (p,q)-equations

AbstractWe consider a nonlinear Robin problem driven by the (p,q)-Laplacian and a parametric reaction exhibiting the competition of a concave term and of a resonant perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameterλmoves on ℝ̊+= (0, +∞). Also, we determine the continuity properties of the solution multifunction.