Thresholded covering algorithms for robust and max–min optimization

In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the \(k\) -robust model where the possible scenarios tomorrow are given by all demand-subsets of size \(k\) . In this paper, we give the following simple and intuitive template for \(k\) -robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for \(k\) -robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our \(k\) -robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, which are based on dual-rounding and primal–dual ideas; these properties might be of some independent interest. As a by-product of our techniques, we also get algorithms for max–min problems of the form: “given a covering problem instance, which \(k\) of the elements are costliest to cover?” For the problems mentioned above, we show that their \(k\) -max–min versions have performance guarantees similar to those for the \(k\) -robust problems.     

Max–max,max–min,min–max and min–min knapsack problems with a parametric constraint

4OR - Max–max, max–min, min–max and min–min optimization problems with a knapsack-type constraint containing a single numerical parameter are studied. The goal is to present...     

Explicit min–max polynomials on the disc

Denote by ${\Pi }_{n+m?1}^2?\{{\sum }_{0\le i+j\le n+m?1}{c}_{i,j}{x}^i{y}^j:c_{i,j}\in \mathbb{R}\}$ the space of polynomials of two variables with real coefficients of total degree less than or equal to $n+m?1$. Let $b_0,b_1,\dots ,b_l\in \mathbb{R}$ be given. For $n,m\in \mathbb{N},n\ge l+1$ we look for the polynomial $b_0{x}^n{y}^m+b_1{x}^{n?1}{y}^{m+1}+?+b_l{x}^{n?l}{y}^{m+l}+q(x,y),q(x,y)\in {\Pi }_{n+m?1}^2$, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial $2P_{n,m}(x,y)=x{G}_{n?1,m}(x,y)+y{G}_{n,m?1}(x,y)=2x^n{y}^m+q(x,y)$ and $q(x,y)\in {\Pi }_{n+m?1}^2$, where $G_{n,m}(x,y)?1/2^{n+m}(U_n\left(x\right)U_m\left(y\right)+U_{n?2}\left(x\right)U_{m?2}\left(y\right))$, and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients $b_j,j=0,\dots ,l,l$ fixed, such that for every $n,m\in \mathbb{N},n\ge l+1$, the linear combination ${\sum }_{\nu =0}^l{b}_{\nu }{P}_{n?\nu ,m+\nu }(x,y)$ is a min–max polynomial. In fact the more general case, when the coefficients $b_j$ and $l$ are allowed to depend on $n$ and $m$, is considered. So far, up to very special cases, min–max polynomials are known only for $x^n{y}^m$,$n,m\in {\mathbb{N}}_0$.     

A metaheuristic for the min–max windy rural postman problem with K vehicles

In this paper we deal with the min–max version of the windy rural postman problem with K vehicles. For this problem, in which the objective is to minimize the length of the longest tour in order to find a set of balanced tours for the vehicles, we present here a metaheuristic that produces very good feasible solutions in reasonable computing times. It is based on the combination of a multi-start procedure with an Iterated Local Search. Extensive computational results on a large set of instances with up to 50 vertices, 184 edges and 5 vehicles are presented. The results are very good, the average gaps with respect to a known lower bound are less than 0.40% for instances with 2 or 3 vehicles and up to 1.60% when 4 or 5 vehicles are considered.     

Semidefinite programming for min–max problems and games

We consider two min–max problems (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies with compact basic semi-algebraic sets of pure strategies. In both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre (SIAM J Optim 11:796–817, 2001; Math Program B 112:65–92, 2008). This provides a unified approach and a class of algorithms to compute Nash equilibria and min–max strategies of several static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation (and sometimes even for finite convergence), a behavior similar to what was observed in polynomial optimization.     

Min–max and min–max (relative) regret approaches to representatives selection problem

The following optimization problem is studied. There are several sets of integer positive numbers whose values are uncertain. The problem is to select one representative of each set such that the sum of the selected numbers is minimum. The uncertainty is modeled by discrete and interval scenarios, and the min?Cmax and min?Cmax (relative) regret approaches are used for making a selection decision. The arising min?Cmax, min?Cmax regret and min?Cmax relative regret optimization problems are shown to be polynomially solvable for interval scenarios. For discrete scenarios, they are proved to be NP-hard in the strong sense if the number of scenarios is part of the input. If it is part of the problem type, then they are NP-hard in the ordinary sense, pseudo-polynomially solvable by a dynamic programming algorithm and possess an FPTAS. This study is motivated by the problem of selecting tools of minimum total cost in the design of a production line.     

Constrained inverse min–max spanning tree problems under the weighted Hamming distance

In this paper, we consider the constrained inverse min–max spanning tree problems under the weighted Hamming distance. Three models are studied: the problem under the bottleneck-type weighted Hamming distance and two mixed types of problems. We present their respective combinatorial algorithms that all run in strongly polynomial times. This research is supported by the National Natural Science Foundation of China (Grant No. 10601051).     

A smoothing algorithm for finite min–max–min problems

We generalize a smoothing algorithm for finite min–max to finite min–max–min problems. We apply a smoothing technique twice, once to eliminate the inner min operator and once to eliminate the max operator. In mini–max problems, where only the max operator is eliminated, the approximation function is decreasing with respect to the smoothing parameter. Such a property is convenient to establish algorithm convergence, but it does not hold when both operators are eliminated. To maintain the desired property, an additional term is added to the approximation. We establish convergence of a steepest descent algorithm and provide a numerical example.     

A min–max vehicle routing problem with split delivery and heterogeneous demand

In this article, we introduce a new variant of min–max vehicle routing problem, where various types of customer demands are satisfied by heterogeneous fleet of vehicles and split delivery of services is allowed. We assume that vehicles may serve one or more types of service with unlimited service capacity, and varying service and transfer speed. A heuristic solution approach is proposed. We report the solutions for several test problems.     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

Generalized skew bisubmodularity: A characterization and a min–max theorem

Huber, Krokhin, and Powell (2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min–max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.     

The Landau–Kolmogorov problem for the Laplace operator on a ball

In this paper we solve the problem of maximizing the value of the Laplace operator at the origin for functions such that the second degree of the Laplace operator belongs to the space L_∞ on the unit ball of the Euclidean space. The problem is solved under restrictions on the uniform norm of a function and the L_∞-norm of the second degree of the Laplace operator of this function.     

Optimal algorithms for the α-neighbor p-center problem

Assigning multiple service facilities to demand points is important when demand points are required to withstand service facility failures. Such failures may result from a multitude of causes, ranging from technical difficulties to natural disasters. The α-neighbor p-center problem deals with locating p service facilities. Each demand point is assigned to its nearest α service facilities, thus it is able to withstand up to α − 1 service facility failures. The objective is to minimize the maximum distance between a demand point and its αth nearest service facility. We present two optimal algorithms for both the continuous and discrete α-neighbor p-center problem. We present experimental results comparing the performance of the two optimal algorithms for α = 2. We also present experimental results showing the performance of the relaxation algorithm for α = 1, 2, 3.     

Inverse Sturm–Liouville problem on equilateral regular tree

In this article, we consider a spectral problem generated by the Sturm–Liouville equation on the edges of an equilateral regular tree. It is assumed that the Dirichlet boundary conditions are imposed at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. The potential in the Sturm–Liouville equations, the same on each edge, is real, symmetric with respect to the middle of an edge and belongs to L ₂(0,?a) where a is the length of an edge. Conditions are obtained on a sequence of real numbers necessary and sufficient to be the spectrum of the considered spectral problem.     

A weighted max–min model for fuzzy goal programming

After Narasimhan's pioneering study of applying fuzzy set theory to goal programming in 1980, many achievements in the field have been recorded. Most of them followed the max–min approach. However, when objectives have different levels of importance, only the weighted additive model of Tiwari et al. seems to be applicable. However, the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave. This study proposes a novel weighted max–min model for fuzzy goal programming (FGP) and for fuzzy multiple objective decision-making. The proposed model adapts well to even the most complicated membership functions. Numerical examples demonstrate that the proposed model can be effectively incorporated with other approaches to FGP and is superior to the weighted additive approach.