Math Oper Res Research Articles

Dynamic Cournot-Nash equilibrium: the non-potential case

We consider a large population dynamic game in discrete time where players are characterized by time-evolving types. It is a natural assumption that the players’ actions cannot anticipate future values of their types. Such games go under the name of dynamic Cournot-Nash equilibria, and were first studied by Acciaio et al. (SIAM J Control Optim 59:2273–2300, 2021), as a time/information dependent version of the games devised by Blanchet and Carlier ( Math Oper Res 41:125–145, 2016) for the static situation, under an extra assumption that the game is of potential type. The latter means that the game can be reduced to the resolution of an auxiliary variational problem. In the present work we study dynamic Cournot-Nash equilibria in their natural generality, namely going beyond the potential case. As a first result, we derive existence and uniqueness of equilibria under suitable assumptions. Second, we study the convergence of the natural fixed-point iterations scheme in the quadratic case. Finally we illustrate the previously mentioned results in a toy model of optimal liquidation with price impact, which is a game of non-potential kind.

Improving the approximation ratio for capacitated vehicle routing

We devise a new approximation algorithm for capacitated vehicle routing. Our algorithm yields a better approximation ratio for general capacitated vehicle routing as well as for the unit-demand case and the splittable variant. Our results hold in arbitrary metric spaces. This is the first improvement upon the classical tour partitioning algorithm by Haimovich and Rinnooy Kan (Math Oper Res 10:527–542, 1985) and Altinkemer and Gavish (Oper Res Lett 6:149–158, 1987).

Splitting games over finite sets

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales \(\{p_t,q_t\}_t\), in order to control a terminal payoff \(u(p_\infty ,q_\infty )\). A first part introduces the notion of “Mertens–Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens–Zamir system for continuous functions on the square \([0,1]^2\). A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237–1257, 2020), we show that the value exists by constructing non Markovian \(\varepsilon \)-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.

Simulated Annealing for Convex Optimization: Rigorous Complexity Analysis and Practical Perspectives

We give a rigorous complexity analysis of the simulated annealing algorithm by Kalai and Vempala (Math Oper Res 31(2):253–266, 2006) using the type of temperature update suggested by Abernethy and Hazan (International Conference on Machine Learning, 2016). The algorithm only assumes a membership oracle of the feasible set, and we prove that it returns a solution in polynomial time which is near-optimal with high probability. Moreover, we propose a number of modifications to improve the practical performance of this method, and present some numerical results for test problems from copositive programming.

Superfast Second-Order Methods for Unconstrained Convex Optimization

In this paper, we present new second-order methods with convergence rate Oleft( k^{-4}right) , where k is the iteration counter. This is faster than the existing lower bound for this type of schemes (Agarwal and Hazan in Proceedings of the 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani and Shiff in Math Program 178(1–2):327–360, 2019), which is Oleft( k^{-7/2} right) . Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from Nesterov (Math Program 186:157–183, 2021) using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relative non-degeneracy condition (Bauschke et al. in Math Oper Res 42:330–348, 2016; Lu et al. in SIOPT 28(1):333–354, 2018). During this process, the Hessian of the objective function is computed once, and the gradient is computed Oleft( ln {1 over epsilon }right) times, where epsilon is the desired accuracy of the solution for our problem.

Pessimism and optimism towards new discoveries

In this paper, we provide an axiomatic foundation of pessimism and optimism towards ambiguity that emerges due to growing awareness. In our setup, this corresponds to a discovery of finer “descriptions” of the original contingencies (states). A decision-maker can form subjective probabilistic beliefs on the original state space and behaves as an expected utility maximizer. However, as finer contingencies are discovered, he may perceive ambiguity with respect to the newly identified states and thus be unable to extend her initial probabilistic beliefs to the expanded state space in an additive way. As a result, the decision-maker’s new beliefs are now “ambiguous” and represented by a probability distribution combined, for each refinement of an original state, with a degree of confidence in this probability estimate. We provide a parametric representation of preferences, identify the DM’s degree of ambiguity as well as his attitude towards ambiguity as captured by the degree of optimism and pessimism. We illustrate the relation of our model to some well-known capacities, such as the E-capacities introduced by Eichberger and Kelsey (Theory Decis 46:107–140, 1999), the JP-capacities of Jaffray and Philippe (Math Oper Res 22:165–185, 1997) and the NEO-additive capacities developed by Chateauneuf et al. (J Econ Theory, 137:538–567, 2007).

Minimization of a class of rare event probabilities and buffered probabilities of exceedance

We consider the problem of choosing design parameters to minimize the probability of an undesired rare event that is described through the average of n i.i.d. random variables. Since the probability of interest for near optimal design parameters is very small, one needs to develop suitable accelerated Monte-Carlo methods for estimating its value. One of the challenges in the study is that simulating from exponential twists of the laws of the summands may be computationally demanding since these transformed laws may be non-standard and intractable. We consider a setting where the summands are given as a nonlinear functional of random variables, the exponential twists of whose distributions take a simpler form than those for the original summands. We use techniques from Dupuis and Wang (Stochastics 76(6):481–508, 2004, Math Oper Res 32(3):723–757, 2007) to identify the appropriate Issacs equations whose subsolutions are used to construct tractable importance sampling (IS) schemes. We also study the closely related problem of estimating buffered probability of exceedance and provide the first rigorous results that relate the asymptotics of buffered probability and that of the ordinary probability under a large deviation scaling. The analogous minimization problem for buffered probability, under conditions, can be formulated as a convex optimization problem. We show that, under conditions, changes of measures that are asymptotically efficient (under the large deviation scaling) for estimating ordinary probability are also asymptotically efficient for estimating the buffered probability of exceedance. We embed the constructed IS scheme in gradient descent algorithms to solve the optimization problems, and illustrate these schemes through computational experiments.

On the Linear Convergence to Weak/Standard d-Stationary Points of DCA-Based Algorithms for Structured Nonsmooth DC Programming

We consider a class of structured nonsmooth difference-of-convex minimization. We allow nonsmoothness in both the convex and concave components in the objective function, with a finite max structure in the concave part. Our focus is on algorithms that compute a (weak or standard) d(irectional)-stationary point as advocated in a recent work of Pang et al. (Math Oper Res 42:95–118, 2017). Our linear convergence results are based on direct generalizations of the assumptions of error bounds and separation of isocost surfaces proposed in the seminal work of Luo and Tseng (Ann Oper Res 46–47:157–178, 1993), as well as one additional assumption of locally linear regularity regarding the intersection of certain stationary sets and dominance regions. An interesting by-product is to present a sharper characterization of the limit set of the basic algorithm proposed by Pang et al., which fits between d-stationarity and global optimality. We also discuss sufficient conditions under which these assumptions hold. Finally, we provide several realistic and nontrivial statistical learning models where all assumptions hold.

Stochastic Dynamic Programming with Non-linear Discounting

In this paper, we study a Markov decision process with a non-linear discount function and with a Borel state space. We define a recursive discounted utility, which resembles non-additive utility functions considered in a number of models in economics. Non-additivity here follows from non-linearity of the discount function. Our study is complementary to the work of Jaśkiewicz et al. (Math Oper Res 38:108–121, 2013), where also non-linear discounting is used in the stochastic setting, but the expectation of utilities aggregated on the space of all histories of the process is applied leading to a non-stationary dynamic programming model. Our aim is to prove that in the recursive discounted utility case the Bellman equation has a solution and there exists an optimal stationary policy for the problem in the infinite time horizon. Our approach includes two cases: (a) when the one-stage utility is bounded on both sides by a weight function multiplied by some positive and negative constants, and (b) when the one-stage utility is unbounded from below.

A value for communication situations with players having different bargaining abilities

The aim of this paper is to extend the Myerson value (Myerson in Math Oper Res 2:225–229, 1977) to situations in which players in a TU-game, in addition to having cooperation possibilities restricted by a graph, also have different bargaining abilities. Then, we will associate to each player in a communication situation a weight in the interval [0, 1] that measures his bargaining ability. A unitary weight corresponds to a fully cooperative player whereas a null weight corresponds to a player that is not willing to cooperate in any way. Intermediate values modulate the bargaining ability. We modify the original TU-game to a new game which is, in turn, a modification of the Myerson’s graph-restricted game. We will assume that the reduction in the will to cooperate implies that players can not obtain the total dividend of the connected coalitions which must be discounted by an appropriate factor. Then, we propose as a solution for these situations the Shapley value (Shapley, in: Kuhn, Tucker (eds) Annals of mathematics studies, Princeton University Press, Princeton, 1953) of the modified game. This solution extends the Myerson value (and also the Shapley value). Moreover it satisfies monotonicity in the weights. Different characterizations of this rule can be obtained. They are based on properties as bargaining component efficiency, fairness, balanced contributions and balanced bargaining ability contributions, and thus they are parallel to those more prominent existing in the literature for the Myerson value.

An Integral Equation Representation for Optimal Retirement Strategies in Portfolio Selection Problem

In this paper we study the consumption and portfolio selection problem of a finitely-lived economic agent with an early retirement option, that is, the agent can choose her/his early retirement time before a mandatory retirement time. Based on the theoretical results in Yang and Koo (Math Oper Res, 43(4):1378–1404, 2018), we derive an integral equation satisfied by the optimal retirement boundary or free boundary using the Mellin transform technique. We also derive integral equation representations for the optimal consumption-portfolio strategies and the optimal wealth process. By using the recursive integration method, we obtain the numerical solutions for the integral equations and discuss economic implications for the optimal retirement strategies by using numerical solutions.

On defining ex ante payoffs in games with diffuse prior

While the diffuse prior has been widely used in applied economic theory for its technical convenience and as a way of modeling complete lack of knowledge, it is not formally defined, nor are ex ante payoffs in games under this prior. In this paper, we provide a formal treatment of the diffuse prior which can validate its application in games. We consider stationary games, in which players’ signals are translation invariant in the true state and players’ payoffs are translation invariant in actions together with the state. We show that strategies which admit well-defined expected payoffs under the diffuse prior are essentially stationary, being almost translation invariant in signals. Our analysis builds on two formal definitions. We define the diffuse prior through a limit construction, using sequences of well-defined priors that become increasingly dispersed. A class of strategy profiles is admissible if for any strategy profile, each player’s ex ante payoff along these sequences converges to a limit that does not depend on the particular sequence. A secondary contribution of the paper is an extension of the concept of distributional strategies (Milgrom and Weber in Math Oper Res 10:619–632, 1985) to a class of multistage games.

On Approximate Stationary Points of the Regularized Mathematical Program with Complementarity Constraints

We discuss the convergence of regularization methods for mathematical programs with complementarity constraints with approximate sequence of stationary points. It is now well accepted in the literature that, under some tailored constraint qualification, the genuine necessary optimality condition for this problem is the M-stationarity condition. It has been pointed out, (Kanzow and Schwartz in Math Oper Res 40(2):253–275. 2015), that relaxation methods with approximate stationary points fail to ensure convergence to M-stationary points. We define a new strong approximate stationarity concept, and we prove that a sequence of strong approximate stationary points always converges to an M-stationary solution. We also prove under weak assumptions the existence of strong approximate stationary points in the neighborhood of an M-stationary solution.

A new notion of error bounds: necessary and sufficient conditions

In this paper, we propose and study a new notion of local error bounds for a convex inequalities system defined in terms of a minimal time function. This notion is called generalized local error bounds with respect to F, where F is a closed convex subset of the Euclidean space $${\mathbb {R}}^n$$ satisfying $$0\in F$$ . It is worth emphasizing that if F is a spherical sector with the apex at the origin then this notion becomes a new type of directional error bounds which is closely related to several directional regularity concepts in Durea et al. (SIAM J Optim 27:1204–1229, 2017), Gfrerer (Set Valued Var Anal 21:151–176, 2013), Ngai and Thera (Math Oper Res 40:969–991, 2015) and Ngai et al. (J Convex Anal 24:417–457, 2017). Furthermore, if F is the closed unit ball in $${\mathbb {R}}^n$$ then the notion of generalized local error bounds with respect to F reduces to the concept of usual local error bounds. In more detail, firstly we establish several necessary conditions for the existence of these generalized local error bounds. Secondly, we show that these necessary conditions become sufficient conditions under various stronger conditions of F. Finally, we state and prove a generalized-invariant-point theorem and then use the obtained result to derive another sufficient condition for the existence of generalized local error bounds with respect to F.

Optimal Strategies in Zero-Sum Repeated Games with Incomplete Information: The Dependent Case

Using the duality techniques introduced by De Meyer (Math Oper Res 21:209–236, 1996a, Math Oper Res 21:237–251, 1996b), Rosenberg (Int J Game Theory 27:577–597, 1998) and De Meyer and Marino (Cahiers de la MSE 27, 2005) provided an explicit construction for optimal strategies in repeated games with incomplete information on both sides, in the independent case. In this note, we extend both the duality techniques and the construction of optimal strategies to the dependent case.

Robust Solutions to the Life-Cycle Consumption Problem

This paper demonstrates how the well-known discrete life-cycle consumption problem (LCP) can be solved using the Robust Counterpart (RC) formulation technique, as defined in Ben-Tal and Nemirovski (Math Oper Res 23(4):769–805, 1998). To do this, we propose a methodology that involves applying a change of variables over the original consumption before deriving the RC. These transformations allow deriving a closed solution to the inner problem, and thus to solve the LCP without facing the curse of dimensionality and without needing to specify the prior distribution for the investment opportunity set. We generalize the methodology and illustrate how it can be used to solve other type of problems. The results show that our methodology enables solving long-term instances of the LCP (30 years). We also show it provides an alternative consumption pattern as to the one provided by a benchmark that uses a dynamic programming approach. Rather than finding a consumption that maximizes the expected lifetime utility, our solution delivers higher utilities for worst-case scenarios of future returns.

Implementable tensor methods in unconstrained convex optimization

In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level Oleft( {1 over k^4}right) , where k is the number of iterations. This is very close to the lower bound of the order Oleft( {1 over k^5}right) , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.

A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs

In this paper, we propose a bounded degree hierarchy of both primal and dual conic programming relaxations involving both semi-definite and second-order cone constraints for solving a nonconvex polynomial optimization problem with a bounded feasible set. This hierarchy makes use of some key aspects of the convergent linear programming relaxations of polynomial optimization problems (Lasserre in Moments, positive polynomials and their applications, World Scientific, Singapore, 2010) associated with Krivine–Stengle’s certificate of positivity in real algebraic geometry and some advantages of the scaled diagonally dominant sum of squares (SDSOS) polynomials (Ahmadi and Hall in Math Oper Res, 2019. https://doi.org/10.1287/moor.2018.0962; Ahmadi and Majumdar in SIAM J Appl Algebra Geom 3:193–230, 2019). We show that the values of both primal and dual relaxations converge to the global optimal value of the original polynomial optimization problem under some technical assumptions. Our hierarchy, which extends the so-called bounded degree Lasserre hierarchy (Lasserre et al. in Eur J Comput Optim 5:87–117, 2017), has a useful feature that the size and the number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. As a special case, we provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for classes of polynomial optimization problems. This includes one-step convergence for a new class of first-order SDSOS-convex polynomial programs. In this case, we also show how a global solution is recovered from the level one SOCP relaxation. We finally derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by concave polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification.

The Myerson Value on Local Structures of Coalitions

The Myerson value introduced by Mayerson (Math Oper Res 2:225–229, 1977) is a solution for cooperative games under the partial cooperation structures described by graphs, in which feasible coalitions are connected but their structures are ignored. To extend the Myerson value, we define a mapping to describe local structures of coalitions and obtain a new solution for cooperative games, called Myerson value with local structures. We propose an axiomatic characterization of the Myerson value associated with local cooperative structures.

The rate of convergence of proximal method of multipliers for equality constrained optimization problems

The proximal method of multipliers was proposed by Rockafellar (Math Oper Res 1:97–116, 1976) for solving convex programming and it is a kind of proximal point method for convex programming. In this paper, we apply this method for equality constrained optimization problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under linear independence constraint qualification and the second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for the equality constrained optimization problem, is linear and the ratio constant is proportional to 1 / c, where c is the penalty parameter that exceeds a threshold $$c_*> 0$$ . Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to $$+\infty $$ .