A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

A unifying theory of exactness of linear penalty functions

In this article, we develop a theory of exact linear penalty functions that generalizes and unifies most of the results on exact penalization existing in the literature. We discuss several approaches to the study of both locally and globally exact linear penalty functions, and obtain various necessary and sufficient conditions for the exactness of a linear penalty function. We pay more attention than usual to necessary conditions, which allows us to deeply understand the exact penalty technique.     

Distributed mirror descent method for saddle point problems over directed graphs

In this article, we consider a mini‐max multi‐agent optimization problem where multiple agents cooperatively optimize a sum of local convex–concave functions, each of which is available to one specific agent in a network. To solve the problem, we propose a distributed optimization method by extending classical mirror descent algorithms to the distributed setting. We obtain the convergence of the algorithm under wild conditions that the agent communication follows a directed graph and the related weighted matrices are row stochastic. In particular, when the weighted matrices are restricted to be doubly stochastic, we provide the explicit convergence rate of the algorithm by choosing the stepsize in a suitable way. The proposed algorithm can be viewed as a generalization of the subgradient projection methods since it utilizes a customized Bregman divergence instead of the usual Euclidean squared distance. Finally, some simulation results on a matrix game are presented to illustrate the performance of the algorithm. © 2016 Wiley Periodicals, Inc. Complexity 21: 178–190, 2016     

Generating functions for computing power indices efficiently

TheShapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. TheBanzhaf power index depends on the number of ways in which each voter can effect a swing. We introduce a combinatorial method based ingenerating functions for computing these power indices efficiently and we study thetime complexity of the algorithms. We also analyze the meet of two weighted voting games. Finally, we compute the voting power in the Council of Ministers of the European Union with the generating functions algorithms and we present its implementation in the system Mathematica. This work has been partially supported by the Spanish Ministery of Science and Technology under grant SEC2000-1243.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

基于二次函数光滑化逼近的修正低阶罚函数

针对不等式约束优化问题, 给出了通过二次函数对低阶精确罚函数进行光滑化逼近的两种函数形式, 得到修正的光滑罚函数. 证明了在一定条件下, 当罚参数充分大, 修正的光滑罚问题的全局最优解是原优化问题的全局最优解. 给出的两个数值例子说明了所提出的光滑化方法的有效性.     

The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

We consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.      

Buyback problem with discrete concave valuation functions

We discuss an online discrete optimization problem called the buyback problem. In the literature of the buyback problem, the valuation function representing the total value of selected elements is given by a linear function. In this paper, we consider a generalization of the buyback problem using nonlinear valuation functions. We propose an online algorithm for the problem with discrete concave valuation functions, and show that it achieves the tight competitive ratio, i.e., the competitive ratio of the proposed algorithm is equal to the known lower bound for the problem.     

一类新的罚函数与罚算法(英)

在本文中,我们提出了带不等式约束的非线性规划问题的一类新的罚函数,它的一个子类可以光滑逼近$l_1$罚函数. 基于此类新的罚函数我们给出了一种罚算法,这个算法的特点是每次迭代求出罚函数的全局精确解或非精确解. 在很弱的条件下算法总是可行的. 我们在不需要任何约束规范的情况下,证明了算法的全局收敛性. 最后给出了数值实验.     

Axiomatizations of Lovász extensions of pseudo-Boolean functions

Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals.     

等式约束优化问题的一类新的简单光滑精确罚函数

精确罚函数方法是求解优化问题的一类经典方法,传统的精确罚函数不可能既是简单的又是光滑的,这里简单的是指罚函数中不包含目标函数和约束函数的梯度信息。针对等式约束问题提出了不同与传统罚函数的一类新的简单光滑罚函数并证明了它是精确的。给出了以新的罚函数为基础的罚函数方法并用数值例子说明算法是可行的。     

Single machine scheduling with exponential learning functions

In this paper we consider the single machine scheduling problem with exponential learning functions. By the exponential learning functions, we mean that the actual job processing time is a function of the total normal processing times of the jobs already processed. We prove that the shortest processing time (SPT) rule is optimal for the total lateness minimization problem. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms. It also shows that the problems of minimizing the total tardiness and discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.     

Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions

We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at the moment. In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one.     

A Gilmore-Gomory construction of integer programming value functions

In this paper, we analyze how sequentially introducing decision variables into an integer program (IP) affects the value function and its level sets. We use a Gilmore-Gomory approach to find parametrized IP value functions over a restricted set of variables. We introduce the notion of maximal connected subsets of level sets - volumes in which changes to the constraint right-hand side have no effect on the value function - and relate these structures to IP value functions and optimal solutions.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

Duality of nonconvex optimization with positively homogeneous functions

We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the p-norm function, where p is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in Mangasarian (Comput Optim Appl 36:43–53, 2007). In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems.     

On the role of continuously differentiable exact penalty functions in constrained global optimization

The aim of this paper is to show that the new continuously differentiable exact penalty functions recently proposed in literature can play an important role in the field of constrained global optimization. In fact they allow us to transfer ideas and results proposed in unconstrained global optimization to the constrained case.First, by drawing our inspiration from the unconstrained case and by using the strong exactness properties of a particular continuously differentiable penalty function, we propose a sufficient condition for a local constrained minimum point to be global.Then we show that every constrained local minimum point satisfying the second order sufficient conditions is an attraction point for a particular implementable minimization algorithm based on the considered penalty function. This result can be used to define new classes of global algorithms for the solution of general constrained global minimization problems. As an example, in this paper we describe a simulated annealing algorithm which produces a sequence of points converging in probability to a global minimum of the original constrained problem.     

An approach for calculating correlation functions in the six-vertex model with domain wall boundary conditions

We address the problem of calculating correlation functions in the six-vertex model with domain wall boundary conditions by considering a particular nonlocal correlation function, called the row configuration probability. This correlation function can be used as a building block for computing various (both local and nonlocal) correlation functions in the model. We calculate the row configuration probability using the quantum inverse scattering method, giving the final result in terms of multiple integrals. We also discuss the relation to the emptiness formation probability, another nonlocal correlation function, which was previously computed using similar methods.     

Restricted generalized Nash equilibria and controlled penalty algorithm

The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem, in which each player’s strategy set may depend on the rival players’ strategies. The GNEP has recently drawn much attention because of its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. However, a GNEP usually has multiple or even infinitely many solutions, and it is not a trivial matter to choose a meaningful solution from those equilibria. The purpose of this paper is two-fold. First we present an incremental penalty method for the broad class of GNEPs and show that it can find a GNE under suitable conditions. Next, we formally define the restricted GNE for the GNEPs with shared constraints and propose a controlled penalty method, which includes the incremental penalty method as a subprocedure, to compute a restricted GNE. Numerical examples are provided to illustrate the proposed approach.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.