A smoothing-out technique for min—max optimization

In this paper, we suggest approximations for smoothing out the kinks caused by the presence of max or min operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR ⁿ . Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.     

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...     

Min–max and min–max regret versions of combinatorial optimization problems: A survey

Min–max and min–max regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the min–max and min–max regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min s–t cut, knapsack. Since most of these problems are NP-hard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.     

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.     

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.     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

Elliptic cone optimization and primal–dual path-following algorithms

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone programmes. We also describe some real-world applications of this class of optimization problems. We study and analyse the Jordan algebraic structure of the elliptic cones. Then, we present a glimpse of the duality theory associated with elliptic cone optimization. A primal–dual path-following interior-point algorithm is derived for elliptic cone optimization problems. We prove the polynomial convergence of the proposed algorithms by showing that the logarithmic barrier is a strongly self-concordant barrier. The numerical examples show the path-following algorithms are efficient.     

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.     

Outer limits of subdifferentials for min–max type functions

Abstract

We generalize the outer subdifferential construction suggested by Cánovas, Henrion, López and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential posed by Li, Meng and Yang.     

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.     

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP). Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum-cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastic speedup in the solution of constrained quadratic 0–1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.     

A generalized min–max theorem for functionals of strongly indefinite sign

We consider non-linear Schrödinger equations of the following type: $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x) + V(x)u(x)-q(x)|u(x)|^\sigma u(x) = \lambda u(x), \quad x\in \mathbb{R }^N \\ u\in H^1(\mathbb{R }^N)\setminus \{0\}, \end{array} \right. \end{aligned}$$ where $N\ge 1$ and $\sigma >0$ . We will concentrate on the case where both $V$ and $q$ are periodic, and we will analyse what happens for different values of $\lambda $ inside a spectral gap $]\lambda ^-,\lambda ^+[$ . We derive both the existence of multiple orbits of solutions and the bifurcation of solutions when $\lambda \nearrow \lambda ^+$ . Thereby we use the corresponding energy function ${I_\lambda }$ and we derive a new variational characterization of multiple critical levels for such functionals: in this way we get multiple orbits of solutions. One main advantage of our new view on some specific critical values $c_0(\lambda )\le c_1(\lambda )\le \cdots \le c_n(\lambda )\le \cdots $ is a multiplicity result telling us something about the number of critical points with energies below $c_n(\lambda )$ , even if for example two of these values $c_i(\lambda )$ and $c_j(\lambda )$ ( $0\le i<j\le n$ ) coincide. Let us close this summary by mentioning another main advantage of our variational characterization of critical levels: we present our result in an abstract setting that is suitable for other problems and we give some hints about such problems (like the case corresponding to a Coulomb potential $V$ ) at the end of the present paper.     

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 $x^n{y}^m$,$n,m\in {\mathbb{N}}_0$.     

Biased random-key genetic algorithms for combinatorial optimization

Random-key genetic algorithms were introduced by Bean (ORSA J. Comput. 6:154–160, 1994) for solving sequencing problems in combinatorial optimization. Since then, they have been extended to handle a wide class of combinatorial optimization problems. This paper presents a tutorial on the implementation and use of biased random-key genetic algorithms for solving combinatorial optimization problems. Biased random-key genetic algorithms are a variant of random-key genetic algorithms, where one of the parents used for mating is biased to be of higher fitness than the other parent. After introducing the basics of biased random-key genetic algorithms, the paper discusses in some detail implementation issues, illustrating the ease in which sequential and parallel heuristics based on biased random-key genetic algorithms can be developed. A survey of applications that have recently appeared in the literature is also given.     

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.