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This paper presents new approximation bounds for trilinear and biquadratic optimization problems over nonconvex constraints. We first consider the partial semidefinite relaxation of the original problem, and show that there is a bounded approximation solution to it. This will be achieved by determining the diameters of certain convex bodies. We then show that there is also a bounded approximation solution to the original problem via extracting the approximation solution of its semidefinite relaxation. Under some conditions, the approximation bounds obtained in this paper improve those in the literature. [ABSTRACT FROM AUTHOR]

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\mathcal {J}(u)=\int_{\partial\Omega}f(x)u\,\mathrm {d}\mathcal {H}^{N-1}$ over some admissible class of loads f where u is the (unique) solution to the problem −Δ p u+| u| p−2 u=0 in Ω with | ∇ u| p−2 u ν= f on ∂Ω. [ABSTRACT FROM AUTHOR]

We found a minor error in the proof of paper “Universal Alignment Probability Revisited” by S.Y. Lin and Y.C. Ho (J. Optim. Theory Appl. 113(2):399–407, ). In this note, we give a counterexample and explain the reason. We also show that the conclusion of that paper is still correct despite this minor error. A new proof of the conclusion is given. [ABSTRACT FROM AUTHOR]

In this paper, we extend Guo and Xia’s necessary condition which has been presented by Guo and Xia (Fuzzy optimizat Decis Mak 5: 33–47, 2006) in order to study the finitely many constraints of fuzzy relation inequalities and optimize a linear objective function on this region which is defined by the fuzzy max–min operator. The new condition provides a means for removing the unnecessary paths resulting from Guo and Xia’s paths. Also, an algorithm and two numerical examples are offered to abbreviate and illustrate the steps of the resolution process of the problem. [ABSTRACT FROM AUTHOR]

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

The order reduction method for singularly perturbed optimal control systems consists of setting the small parameter equal to zero and employing the differential system thus obtained. Although in many situations this provides the correct variational limit problem, it is established in this paper that when considering systems with non-scalar fast variables, the set of systems for which the order reduction method is invalid is dense in the class of systems under consideration. This extends previous results, where only systems with linear fast variables were considered. The present result complements a result established in a joint work with Artstein, where it was established that the order reduction method is valid for singularly perturbed optimal control systems with scalar fast variable. [ABSTRACT FROM AUTHOR]

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problems coincide. [ABSTRACT FROM AUTHOR]

In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved. [ABSTRACT FROM AUTHOR]

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda>0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p<+\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings. [ABSTRACT FROM AUTHOR]

degenerate parabolic system of equations of two-phase multicomponent filtration is considered. It is shown that this system can be treated as a system of conservation laws and the notions developed in the corresponding theory, such as hyperbolicity, shock waves, Hugoniot relations, stability conditions, Riemann problem, entropy, etc., can be applied to this system. The specific character of the use of such notions in the case of multicomponent filtration is demonstrated. An example of two-component mixture is used to describe the specific properties of solutions of the Riemann problem. [ABSTRACT FROM AUTHOR]

We study multiobjective optimization problems with equilibrium constraints (MOPECs) described by parametric generalized equations in the form 0 ϵ G(x, y) + Q(x, y), where both mappings G and Q are set-valued. Such models arise particularly from certain optimization-related problems governed by variational inequalities and firstorder optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications by using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex while nondifferentiable data. [ABSTRACT FROM AUTHOR]

An alternative technique for recovering the technology from the cost function is investigated. The recovery method uses an optimization problem with a finite number of constraints that, under appropriate conditions, has the equations of Shephard's lemma among the first order conditions. Hence, a justification is provided for the technique of recovering the production function from the cost function by solving the system of equations generated by Shephard's lemma. The alternative technique also recovers the output distance function from the cost function in the multioutput case. [ABSTRACT FROM AUTHOR]

In this note, it is proved for n ≤ 5 that if Q is a finite quantale with | Q| ≥ n, then there are at least n subquantales of Q. However, the result does not hold when | Q| ≥ 6. Also, an example is given of a sequence of quantales of cardinality p + 2 for p prime, each of which have exactly 5 subquantales. [ABSTRACT FROM AUTHOR]

We consider the control problem for a group containing a given number of uninformed agents; this is a consensus problem. An agent's model is defined by a two-dimensional integrator. The purpose of this work is to synthesize a decentralized smooth law of the agents' motion in a multiagent system that would let us construct and support a prescribed configuration of the group of agents. The considered group should follow the leader, either real or virtual. Informational links between agents are given by a communications digraph. The work shows numerical results for various initial conditions and various desired trajectories. [ABSTRACT FROM AUTHOR]

We determine necessary conditions under which integrable vector evolution equations of third order admit Miura-type transformations. For equations integrable on the n-dimensional sphere, we obtain first-order differential substitutions. [ABSTRACT FROM AUTHOR]

Zero-sum ergodic semi-Markov games with weakly continuous transition probabilities and lower semicontinuous, possibly unbounded, payoff functions are studied. Two payoff criteria are considered: the ratio average and the time average. The main result concerns the existence of a lower semicontinuous solution to the optimality equation and its proof is based on a fixed-point argument. Moreover, it is shown that the ratio average as well as the time average payoff stochastic games have the same value. In addition, one player possesses an ε-optimal stationary strategy ( ε>0), whereas the other has an optimal stationary strategy. [ABSTRACT FROM AUTHOR]

A mixed optimal model approximation is presented to obtain reduced-order models for truly fast descriptor systems. By a projection from truly fast descriptor systems to discrete-time systems, a mixed optimal model approximation for truly fast descriptor systems is transformed to a mixed optimal model approximation of the corresponding discrete-time systems. The structure of the fast descriptor systems is preserved in the model approximation procedure. The expression of the error and its gradient are given explicitly in terms of the solutions of certain Lyapunov equations. A numerical example is provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We study an optimization problem for an elliptic PDE where the unknown is a measure. The state equation is a relaxed form of the classical Dirichlet problem and the functional to minimize is nonlinear. We prove a regularity result for the optimal measure, give a description of it and present some numerical examples. [ABSTRACT FROM AUTHOR]

Necessary conditions are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The structure of the optimal control problem under consideration is exploited and special emphasis is laid on the representation of the Lagrange multipliers resulting from the necessary conditions for infinite optimization problems. [ABSTRACT FROM AUTHOR]

Classical shallow-ice theory assumes that bed topography under ice sheets has slopes comparable to the surface slope of the ice sheet. A modification of the classical steady-state theory which allows for significant bed topography on shorter length scales has recently been developed by Morland (Proc. R. Soc L. Ser. A., 456, 1711-1739), but his theory requires explicit integration of the ice-flow equations over the topography length scale, which may be below the grid size of typical numerical ice sheet models. Here we present a method for parameterising the effect of basal topography of wavelengths much greater than ice thickness but much smaller than the horizontal extent of the ice sheet on the bulk flow of the ice sheet. In particular, we are able to show through the use of a multiple-scales expansion technique that the effect of such topography is described by a simple correction factor applied to the classical expression for ice flux. This correction factor dispenses with the need to integrate explicitly over the topography length scale and could allow the effect of such topography to be included in numerical models with limited grid size. Examples are given for the practical implementation of this 'correction factor method' in calculations of the steady-state shape of ice sheets. [ABSTRACT FROM AUTHOR]

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, for stress-free and rigid-rigid boundary conditions and Schmidt numbers PC greater than Prandtl numbers PT , is studied in the region around the bifurcation point &scriptC;02 = &scriptR;B2 (PT +1)/[PT (p-1)] of linear instability.An improvement of the results in Mulone [11] is found for small values of p = PC/PT and PT. For p suf.ciently large the critical nonlinear Rayleigh number is very close to the linear one (with relative difference less than 1% in the sea water case) [ABSTRACT FROM AUTHOR]

In this paper we study fair division problems with the special feature that there exists only one transferable good that everyone likes. This good will be used to compensate some individuals for their differences in other non-transferable resources (like talents or handicaps). In this context we test the traditional no-envy solution and we verify that: 1) its ethical content can be a matter of discussion, and 2) frequently it does not select a non-empty set of allocations. We propose an extension of this criterion that partially solves the existence problem while also retaining the main ethical properties of the preceding solution. [ABSTRACT FROM AUTHOR]