Lipschitz条件下的加权梯形不等式和中点不等式

利用与一阶导数有关的积分恒等式,并通过引入参数求最值,在一 阶导函数满足Lipschitz条件的情况下,给出加权梯形不等式和中点不等式.     

Boundedness and Regularity Properties of Semismooth Reformulations of Variational Inequalities

The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only. This work is supported by the Research Grant Council of Hong Kong This paper is dedicated to Alex Rubinov on the occasion of his 65th Birthday     

MM algorithms for geometric and signomial programming

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.     

New perspective on the Kuhn–Tucker theorem

A new proof of the Kuhn–Tucker theorem on necessary conditions for a minimum of a differentiable function of several variables in the case of inequality constraints is given. The proof relies on a simple inequality (common in textbooks) for the projection of a vector onto a convex set.     

On the effect of major vertices on the number of light edges

This paper presents an inequality satisfied by planar graphs of minimum degree five. For the purposes of this paper, an edge of a graph is light if the weight of the edge, or the sum of the degrees of the vertices incident with it, is at most eleven. The inequality presented shows that planar graphs of minimum degree five have a large number of light edges. This inequality improves upon a recent inequality of Borodin and Sanders, which showed that 7/15 times the number of edges of weight 10 plus 1/5 times the number of edges of weight 11 is at least 12. These constants 7/15 and 1/5 were shown to be best possible. The inequality in this paper shows that, for this type of graph, the presence of vertices of degree at least eight increases the number of light edges. A graph is presented which shows that the coefficient obtained for the number of degree eight vertices is best possible. © 1996 John Wiley & Sons, Inc.     

Image Space Analysis for Vector Variational Inequalities with Matrix Inequality Constraints and Applications

In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.     

Dual Sufficient Optimality Conditions for the Generalized Problem of Bolza

We develop sufficient conditions for optimality in the generalized problem of Bolza. The basis of our approach is the dual Hamilton–Jacobi inequality leading to a new sufficient criterion for optimality in which we assume the existence of a function satisfying, together with the Hamiltonian, a certain inequality. Consequently, using this criterion, we derive new sufficient conditions for optimality of first and second order for a relative minimum.     

Direct solution to constrained tropical optimization problems with application to project scheduling

We examine a new optimization problem formulated in the tropical mathematics setting as a further extension of certain known problems. The problem is to minimize a nonlinear objective function, which is defined on vectors over an idempotent semifield by using multiplicative conjugate transposition, subject to inequality constraints. As compared to the known problems, the new one has a more general objective function and additional constraints. We provide a complete solution in an explicit form to the problem by using an approach that introduces an auxiliary variable to represent the values of the objective function, and then reduces the initial problem to a parametrized vector inequality. The minimum of the objective function is evaluated by applying the existence conditions for the solution of this inequality. A complete solution to the problem is given by solving the parametrized inequality, provided the parameter is set to the minimum value. As a consequence, we obtain solutions to new special cases of the general problem. To illustrate the application of the results, we solve a real-world problem drawn from time-constrained project scheduling, and offer a representative numerical example.     

Some Remarks On Vector Optimization Problems

We prove the existence of a weak minimum for vector optimization problems by means of a vector variational-like inequality and preinvex mappings.     

Some Applications of a Polynomial Inequality to Global Optimization

In this paper, we use the Ehlich-Zeller-Gärtel inequality to derive an algorithm for finding the global minima of polynomials over hyperrectangles as well as to provide a bounding method for the branch-and-bound algorithm. The latter application of the inequality results in an improved algorithm which gives simultaneously a decreasing upper bound and an increasing lower bound for the global minimum at each iteration. The algorithm can be used also to find the Lipschitz constant of a polynomial.     

An exact penalty function for nonlinear programming with inequalities

It is shown how, given a nonlinear programming problem with inequality constraints, it is possible to construct an exact penalty function with a local unconstrained minimum at any local minimum of the constrained problem. The unconstrained minimum is sufficiently smooth to permit conventional optimization techniques to be used to locate it. Numerical evidence is presented on five well-known test problems.     

Existence of solutions for vector optimization

In this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions.     

On the Definition of a Minimum in Parameter Optimization

For the unconstrained minimization of an ordinary function, there are essentially two definitions of a minimum. The first involves the inequality \(\le \), and the second, the inequality <. The purpose of this Forum is to discuss the consequences of using these definitions for finding local and global minima of the constant objective function. The first definition says that every point on the constant function is a local minimum and maximum, as well as a global minimum and maximum. This is not a rational result. On the other hand, the second definition says that the constant function cannot be minimized in an unconstrained problem. It must be treated as a constrained problem where the constant function is the lower boundary of the feasible region. This is a rational result. As a consequence, it is recommended that the standard definition (\(\le \)) for a minimum be replaced by the second definition (<).     

On optimality conditions in nonsmooth inequality constrained minimization

First order necessary optimality conditions for a minimum of an inequality constrained minimization problem are given in terms of approximate quasidifferentials, without the usual differentiability, convexity or locally Lipschitz assumptions. The main result is obtained with the help of a semi-infinite Gordan type alternative theorem. Sufficient conditions for a minimum are also given with the usual convexity assumption replaced by an invex condition.     

Necessary and Sufficient Conditions for Efficiency Via Convexificators

Based on the extended Ljusternik Theorem by Jiménez-Novo, necessary conditions for weak Pareto minimum of multiobjective programming problems involving inequality, equality and set constraints in terms of convexificators are established. Under assumptions on generalized convexity, necessary conditions for weak Pareto minimum become sufficient conditions.     

An application of Guignard's generalized Kuhn-Tucker conditions

Necessary conditions are given for a real-valued function to have a minimum subject to an inequality constraint. Under the appropriate hypotheses, the problem is demonstrated to be a special case of the type of problem to which Guignard's Kuhn-Tucker theorem can be applied.     

A high-dimensional Wilks phenomenon

A theorem by Wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a Chi-square distribution where the number of degrees of freedom coincides with the model dimension. This observation is at the core of some goodness-of-fit testing procedures and of some classical model selection methods. This paper describes a non-asymptotic version of the Wilks phenomenon in bounded contrast optimization procedures. Using concentration inequalities for general functions of independent random variables, it proves that in bounded contrast minimization (as for example in Statistical Learning Theory), the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk (the excess empirical risk) satisfies a Bernstein-like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions. From a mathematical statistics viewpoint, the significance of this result comes from the recent observation that when using model selection via penalization, the excess empirical risk represents a minimum penalty if non-asymptotic guarantees concerning prediction error are to be provided. From the perspective of empirical process theory, this paper describes a concentration inequality for the supremum of a bounded non-centered (actually non-positive) empirical process. Combining the now classical analysis of M-estimation (building on Talagrand??s inequality for suprema of empirical processes) and versatile moment inequalities for functions of independent random variables, this paper develops a genuine Bernstein-like inequality that seems beyond the reach of traditional tools.     

Existence of Solutions for Vector Optimization on Hadamard Manifolds

In this paper, a relationship between a vector variational inequality and a vector optimization problem is given on a Hadamard manifold. An existence of a weak minimum for a constrained vector optimization problem is established by an analogous to KKM lemma on a Hadamard manifold.     

Meritocratic and monopoly inequality: A computer simulation of income distribution

This paper explores the properties of a model of the distribution of income in which individual income is proportional to a multiplicative function of previous income, ability, chance, a ceiling factor determined by competition among members of an income class for resources held by members of other classes, and an additive factor summarizing effects of altruism and minimal subsistence. The behavior of the model is investigated by computer simulation for combinations of values of three model parameters representing the tendency of income to grow exponentially (the Monopoly effect), the weight of the ability factor (the meritocracy effect), and the weight of the ceiling factor resulting from competitive interactions. Steady state income distributions generated by the model are characterized by measures of income inequality, exchange mobility, elite stability, and meritocracy. Results suggest that for constant Monopoly effect, the effect of the meritocracy parameter on various aggregate outcomes is nonlinear, with a range over which greater returns to ability produce lower inequality, lower exchange mobility, greater elite stability and meritocracy, for constant returns to ability, a greater Monopoly effect generally produces greater inequality, more exchange mobility, less stability of the elite, and lower meritocracy. Results also reveal a nonlinear relationship between exchange mobility and inequality, with mobility decreasing to a minimum and then increasing again as inequality increases; a nonlinear but monotonic negative relationship between elite stability and inequality, with greater inequality, associated with less stability, and a nonlinear relationship between meritocracy and inequality, with meritocracy increasing at first with inequality at low inequality levels, reaching a maximum and then decreasing as inequality increases further. These findings are interpreted in relation to major stratification trends in the course of sociocultural evolution.     

On the relative extrema of a linear combination of elementary symmetric functions

We determine where a linear combination of elementary symmetric functions attains as maximum and minimum over a certain convex set in Rⁿ. We also show that an inequality for elementary symmetric functions proposed by S. Pierce is true.