Lagrange Multiplier Rules for Weak Approximate Pareto Solutions of Constrained Vector Optimization Problems in Hilbert Spaces

In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886–911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.     

Flows on open manifolds with positively Lagrange stable trajectories

We give the conditions for a flow generated by a smooth vector field X which guarantee that every smooth vectorfield Y in some C⁰-neighborhood of X defines a flow with positively Lagrange stable trajectories.     

Discrete Lagrange problems with constraints valued in a Lie group

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group $SO(n)$.     

Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps

This paper extends the concept of cone subconvexlikeness of single-valued maps to set-valued maps and presents several equivalent characterizations and an alternative theorem for cone-subconvexlike set-valued maps. The concept and results are then applied to study the Benson proper efficiency for a vector optimization problem with set-valued maps in topological vector spaces. Two scalarization theorems and two Lagrange multiplier theorems are established. After introducing the new concept of proper saddle point for an appropriate set-valued Lagrange map, we use it to characterize the Benson proper efficiency. Lagrange duality theorems are also obtained     

生成锥内部凸-锥-类凸集值优化问题的Henig真有效性

该文讨论局部凸空间中的约束集值优化问题. 首先, 在生成锥内部凸-锥-类凸假设下, 建立了Henig真有效解在标量化和Lagrange乘子意义下的最优性条件. 其次, 对集值Lagrange映射引入Henig真鞍点的概念, 并用这一概念刻画了Henig真有效解. 最后, 引入了一个标量Lagrange对偶模型, 并得到了关于Henig真有效解的对偶定理. 另外, 该文所得结果均不需要约束序锥有非空的内部.     

Hartley Proper Efficiency in Multiobjective Optimization Problems with Locally Lipschitz Set-valued Objectives and Constraints

In this paper we give necessary conditions for Hartley proper efficiency in a vector optimization problem whose objectives and constraints are described by nonconvex locally Lipchitz set-valued maps. The obtained necessary conditions are written in terms of a Lagrange multiplier rule. Our approach is based on a reduction theorem which leads the problem of studying proper efficiency to a scalar optimization problem whose objective is given by a function of max-type. Sufficient conditions for Hartley proper efficiency are also considered.     

The Euler-Lagrange method in Pontryagin’s formulation

The classical variational problem with nonholonomic constraints is solvable by the Euler-Lagrange method in Pontryagin’s formulation; however, in this case Lagrange multipliers are merely measurable functions. In this paper, we put forward a modified Euler-Lagrange method, in which the original problem involves a Lagrangian dependent only on the independent components of the velocity vector. Under this approach, the Lagrange multipliers make up an absolutely continuous vector function. Our method is applied to the problem of horizontal geodesics for a nonholonomic distribution on a manifold. These equations are established as having two types of connections: connection on the distribution and connection on the manifold; this was not accounted for by other researchers.     

Quadratic optimization problems in robust beamforming

This paper presents a class of constrained optimization problems whereby a quadratic cost function is to be minimized with respect to a weight vector subject to an inequality quadratic constraint on the weight vector. This class of constrained optimization problems arises as a result of a motivation for designing robust antenna array processors in the field of adaptive array processing. The constrained optimization problem is first solved by using the primal-dual method. Numerical techniques are presented to reduce the computational complexity of determining the optimal Lagrange multiplier and hence the optimal weight vector. Subsequently, a set of linear constraints or at most linear plus norm constraints are developed for approximating the performance achievable with the quadratic constraint. The use of linear constraints is very attractive, since they reduce the computational burden required to determine the optimal weight vector.     

Ample and spanned vector bundles of top Chern number two on smooth projective varieties

The purpose of this paper is to classify ample and spanned vector bundles of top Chern number two on smooth projective varieties of arbitrary dimension defined over an algebraically closed field of characteristic zero.

     

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

广义锥-s次类凸向量优化的标量化和鞍点定理

讨论序拓扑向量空间中的约束向量优化问题.在广义锥-s次类凸假设下,得到了向量优化问题关于δ-弱有效解的标量化定理和Lagrange泛函的鞍点定理.     

Superefficiency in Vector Optimization with Nearly Subconvexlike Set-Valued Maps

In the framework of locally convex topological vector spaces, we establish a scalarization theorem, a Lagrange multiplier theorem and duality theorems for superefficiency in vector optimization involving nearly subconvexlike set-valued maps.     

Spin Stability of a Lagrange Top Containing Linear Oscillators

The author estimates the influence of movable point masses (linear oscillators) vibrating along the symmetry axis of a top or along axes orthogonal to the symmetry axis on the stability of uniform spinning of a Lagrange top.     

Polarization of the Electron–Positron Vacuum by a Strong Magnetic Field in the Theory with a Fundamental Mass

In the framework of the theory with a fundamental mass in the one-loop approximation, we evaluate the exact Lagrange function of the strong constant magnetic field, replacing the Heisenberg–Euler Lagrangian in the traditional QED. We establish that the derived generalization of the Lagrange function is real for arbitrary values of the magnetic field. In the weak field, the evaluated Lagrangian coincides with the known Heisenberg–Euler formula. In extremely strong fields, the field dependence of the Lagrangian completely disappears; in this range, the Lagrangian tends to the limit value determined by the ratio of the fundamental and lepton masses.     

有关中值定理的探究性教学

通过对中值定理教学思路的设计,给出探究性教学方法的一个实例,即通过导数概念的物理意义导出Lagrange中值定理,经特殊化后推出Rolle定理,再经化归思想给出Lagrange定理的证明,最后推广得到Cauchy中值定理,并借助类比或化归思想分别给出Cauchy定理的证明．     

Characterizations of solution sets of convex vector minimization problems

Complete dual characterizations of the weak and proper optimal solution sets of an infinite dimensional convex vector minimization problem are given. The results are expressed in terms of subgradients, Lagrange multipliers and epigraphs of conjugate functions. A dual condition characterizing the containment of a closed convex set, defined by a cone-convex inequality, in a reverse-convex set, plays a key role in deriving the results. Simple Lagrange multiplier characterizations of the solution sets are also derived under a regularity condition. Numerical examples are given to illustrate the significance of the results.     

Lagrange Multipliers for <Emphasis Type="Italic">ε</Emphasis>-Pareto Solutions in Vector Optimization with Nonsolid Cones in Banach Spaces

This paper presents some results concerning the existence of the Lagrange multipliers for vector optimization problems in the case where the ordering cone in the codomain has an empty interior. The main tool for deriving our assertions is a scalarization by means of a functional introduced by Hiriart-Urruty (Math. Oper. Res. 4:79–97, 1979) (the so-called oriented distance function). Moreover, we explain some applications of our results to a vector equilibrium problem, to a vector control-approximation problem and to an unconstrainted vector fractional programming problem.     

Higher Order Efficiency,Saddle Point Optimality,and Duality for Vector Optimization Problems

In this paper, we intend to characterize the strict local efficient solution of order m for a vector minimization problem in terms of the vector saddle point. A new notion of strict local saddle point of higher order of the vector-valued Lagrangian function is introduced. The relationship between strict local saddle point and strict local efficient solution is derived. Lagrange duality is formulated, and duality results are presented.     

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates. Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999     

Abel’s theorem and Bäcklund transformations for the Hamilton-Jacobi equations

We consider an algorithm for constructing auto-Bäcklund transformations for finitedimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Bäcklund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Bäcklund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane.