In practice, min-maxproblems are often solved us- ing gradient-based algorithms, especially simultaneous gradient descent ascent (GDA) that simply alternates between a gradient descent step for x and a gradi- ent ascent step for y. While these algorithms are attractive due to their simplicity, there are however cases where the gradient of the objective function is not accessible, such as when modelling distributions with categorical variables (Jang et al., 2016), tuning hyper-parameters (Audet and Orban, 2006; Marzat et al., 2011) and multi-agent reinforcement learning with bandit feedback (Zhang et al., 2019). A resur- gence of interest has recently emerged for applications in black-box optimization (Bogunovic et al., 2018; Liu et al., 2019) and black-box poisoning attack (Liu et al., 2020), where an attacker deliberately modifies the training data in order to tamper with the model’s predictions. This can be formulated as a min-max optimization problem, where only stochastic accesses to the objective function are available (Wang et al., 2020).
We study the ridge method for min-maxproblems, and investigate its conver- gence without any convexity, differentiability or qualification assumption. The cen- tral issue is to determine whether the “parametric optimality formula” provides a conservative field, a notion of generalized derivative well suited for optimization. The answer to this question is positive in a semi-algebraic, and more generally de- finable, context. The proof involves a new characterization of definable conservative fields which is of independent interest. As a consequence, the ridge method applied to definable objectives is proved to have a minimizing behavior and to converge to a set of equilibria which satisfy an optimality condition. Definability is key to our proof: we show that for a more general class of nonsmooth functions, conservativity of the parametric optimality formula may fail, resulting in an absurd behavior of the ridge method.
R. LARAKI AND J.B. LASSERRE
Abstract. We consider two min-maxproblems: (1) minimizing the supre- mum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strate- gies 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 [24, 26]. 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.
The next lemma shows that the running times of our distributed algorithms are asymptotically tight, while the ap- proximation factor cannot be improved without introducing a linear dependency on the diameter of the underlying graph. Lemma III.13 (Lower Bound on the Running time). For γ ≥ 2, any distributed algorithm approximating the coreness values, the maximal densities or the min-max edge orientation problem with an approximation ratio strictly smaller than γ requires Ω( log n log γ ) communication rounds, where n is the number of nodes in the underlying graph (n sufficiently large). Moreover, any distributed algorithm for the aforementioned problems with an approximation ratio strictly smaller than 2 would require Ω(n) communication rounds.
MARIA J. ESTEBAN, MATHIEU LEWIN, AND ´ ERIC S´ ER ´ E
Abstract. This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution µ. We show here that the operator has a unique distinguished self-adjoint extension under the sole condi- tion that µ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain us- ing a quadratic form associated with the upper spinor, following earlier works [15, 16] by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigen- values have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac-Coulomb operators corresponding to µ being a finite sum of deltas.
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ABSTRACT: We deal in this paper with SDRALBP-2, namely the Sequence-Dependent Robotic Assembly Line Balancing Problem of type 2. The problem is of industrial relevance due to the growing robotization of the assembly lines in the new Industry 4.0 era. Given a set of operations that are necessary to assembly a product and a set of robot types with different performances, the problem is concerned with addressing three decision problems simultaneously while minimizing a given objective. The first decision is to assign the operations to a given set of stations placed in a straight line [Balancing decision], the second decision is to sequence the operations in each station due the sequence-dependent setup times [Sequencing decision] and the third decision is to assign a robot to each station [Equipment selection decision]. We consider the objective of minimizing the cycle time, which is the maximum duration spent by a product in some station. We propose in this paper a method of type Sequence-First Balance-And-Select-Second. The proposed method embeds a dynamic programming algorithm (that solves a polynomial case) in a metaheuristic. Benchmark instances are used to evaluate the proposed method.
For the example presented in this section, we consider a FSR equal to 0.8 nm and, as mentioned before, an optimal repartition of wavelengths onto this interval.
Fig. 3. Execution times comparison for min vs max wavelength allocation. The Fig. 3 and Table I highlight the opportunity of wave- length allocation. In particular, the column “gain” of table I shows the reduction of execution time when the number for wavelengths in the waveguide increases. For example, for the Appli 1 and for 16 wavelengths in the waveguide, the possibility to allocate 16 wavelengths for each communication leads to an execution time reduction from 73 to 48, i.e. which corresponds to 34.2% of reduction. For the example ”Appli 5”, which corresponds to a larger task graphs with more communications between tasks, the possibility to support greater wavelengths allocation leads to an important reduction in terms of the execution time of the application (from 147 to 85, i.e 42% of execution time reduction).
Keywords: min-max, min-max regret, computational complexity, pseudo-polynomial.
The definition of an instance of a combinatorial optimization problem requires to specify parameters, in particular objective function coefficients, which may be uncertain or imprecise. Uncertainty/imprecision can be structured through the concept of scenario which corresponds to an assignment of plausible values to model parameters. Each scenario s can be represented as a vector in IR m where m is the number of relevant numerical parameters. Kouvelis and Yu  proposed the maximum cost and maximum regret criteria, stemming from decision theory, to construct solutions hedging against parameters variations. In min-max optimization, the aim is to find a solution having the best worst case value across all scenarios. In min-max regret problem, it is required to find a feasible solution minimizing the maximum deviation, over all possible scenarios, of the value of the solution from the optimal value of the corresponding scenario. Two natural ways of describing the set of all possible scenarios S have been considered in the literature. In the interval data case, each numerical parameter can take any value between a lower and upper bound, independently of the values of the other parameters. Thus, in this case, S is the cartesian product of the intervals of uncertainty for the parameters. In the discrete scenario case, S is described explicitly by the list of all vectors s ∈ S. In this case, that is considered in this paper, we distinguish situations where the number of scenarios is bounded by a constant from those where the number of scenarios is unbounded.
could decide if I contains a feasible partition (we could even derive a maximum constrained partition by selecting for A ′ the set of items present in the optimal
We conclude this section giving some precisions about the complexity status of these problems. Pseudo-polynomial time algorithms were given in the case of a bounded number of scenarios for min-max (max-min) and min-max regret versions of shortest path, knapsack, and minimum spanning tree on grid graphs . Our fptas for min-max and min-max regret spanning tree establish also the existence of pseudo-polynomial time algorithms for these problems. Thus min-max (max-min) and min-max regret versions of shortest path, minimum spanning tree and knapsack are weakly NP -hard.
1.1 Research objectives
The purpose of the thesis is to develop abstract models for both the DCDC and DCDCA problems. The former is formulated as a dynamic game involving a large number of agents initially spread out in an Euclidean space. These agents must move within a finite time horizon from their initial positions towards one of multiple predefined alternatives. They do so while exerting as little effort as possible, and trying to remain grouped around their average. According to whether they act selfishly or cooperate, the DCDC problem is modeled as respectively a dynamic non-cooperative game or a social optimization problem. The DCDCA problem is, however, formulated by adding an extra “major” agent (Huang, 2010) (called advertiser in this case) to the non-cooperative DCDC game with two alternatives. The advertiser makes some investments to subsequently influence the paths of the other “minor” agents (called consumers in this case) and persuade them to choose a specific alternative among the two. Within this framework, our work aims at answering the following questions: • What are the optimal strategies that lead the agents to settle with least effort on one
00(u 0 ,u 1 )
LD (F )
4.3 Comparing the bounds
The CGRL algorithm proposed in  (initially introduced in ) for addressing the minmax prob- lem uses the procedure described in  for computing a lower bound on the return of a policy given a sample of trajectories. More specifically, for a given sequence (u 0 , u 1 ) ∈ U 2 , the program
1. The notation OftLess stresses the fact that f is often much smaller than g : consider functions φ which are much smaller than the identity function, e.g. max(0, z − c), ⌊z/c⌋, ⌊log(z)⌋, log ∗ (z),. . .
2. OftLess C,D F carries the contents, reformulated in terms of uniform con- structive (C, D)-density, of Barzdins result cited above, and that of adequate variants that we shall prove about K max and K min (cf. Lemmas 8.1, 8.6, 8.8).
esprit, saisi par la grâce, Fra Angelico a accompli, à l’envers en quelque sorte, le miracle de
l’incarnation puisque, sous son pinceau, les habitants du paradis, la Vierge Marie, Saint Dominique, voire le Christ, lui-même retrouvent une vie terrestre. Jacob a saisi la spécificité de l’artiste florentin qui s’écarte d’une esthétique naturaliste. En lui, l’homme, l’artiste et le saint ne forment plus qu’un seul être. Et, comme par surcroît, ses œuvres accomplissent le miracle de convertir en silence leurs spectateurs (on connaît le zèle prosélyte du poète). Enfin, parce qu’il a vécu à une époque où la figure de l’artiste commence à prendre forme, Fra Angelico a gagné cette gloire qui tente toujours Max Jacob, même s’il y renonce.
Il n’est donc pas inintéressant de caractériser Max en tant que langage de programmation : il faudrait alors le considérer comme un langage impératif, du fait de l’im- portance de la séquentialité et des états . En fait, un usage soigneux des objets-variables et du contrôle séquen- tiel (à travers les objets mentionnés plus haut et d’autres encore) permet un style de programmation manifeste- ment impératif, qui a l’avantage de produire des patches « étiquetés » et clairement structurés (voir figure 2). Ce style n’est d’ailleurs pas toujours pratique, puisqu’il tend à produire des patches beaucoup plus grands que leurs équivalents qui profitent des nombreux « raccourcis » of- ferts par Max. Ces raccourcis, malheureusement, ont ten- dance à engendrer des graphes compliqués, dans lesquels les données suivent des parcours très ramifiés, de type « spaghetti » , qui distribuent l’état du programme à travers de nombreux objets localisés dans de nombreux patches (voir figure 3).
For the class of systems comprising differential inclusions, and state-dependent switched systems, we in- troduced a family of nonsmooth functions obtained by max-min combinations. Based on two notions of generalized directional derivatives, we proposed sufficient conditions for global asymptotic stability. For a class of systems with conic switching regions and linear dynamics within each of these regions, we studied some conditions under which a max-min condition can be obtained by solving matrix inequalities. A possible route for future research is the generalization of this approach to a wider class of systems, and develop further numerical tools for checking the proposed Lie derivative based conditions.
In the present context of i) growing demand, not only for base metals but also for “new“ commodities and materials used in the high-tech sector, including green technologies; ii) anticipated decrease in the use of fossil energy driven by environmental concerns if not by supply problems; and iii) increase of world population and anthropogenic environmental footprint, the EU faces a range of critical issues that can threaten the security of supply of the highly diversified mineral resources needed by its economy. Securing supplies has therefore become crucial. This requires advanced research and innovation to improve the capacity of existing technologies to discover new deposits, and to improve the efficiency of the entire geomaterials life cycle, from mineral extraction and processing to product design, use, reuse and the exploitation as secondary resource of products at the end of their industrial life. Closing the loop, the development of a circular economy incorporating a maximum level of recycling, substitution and optimised use of resources must become a top priority if the challenges faced by humanity in the coming decades are to be addressed.