Primal-dual Structure Research Articles

A Multi-Population Mean-Field Game Approach for Large-Scale Agents Cooperative Attack-Defense Evolution in High-Dimensional Environments

As one of the important issues of multi-agent collaboration, the large-scale agents’ cooperative attack–defense evolution requires a large number of agents to make stress-effective strategies to achieve their goals in complex environments. Multi-agent attack and defense in high-dimensional environments (3D obstacle scenarios) present the challenge of being able to accurately control high-dimensional state quantities. Moreover, the large scale makes the dynamic interactions in the attack and defense problems increase dramatically, which, using traditional optimal control techniques, can cause a dimensional explosion. How to model and solve the cooperative attack–defense evolution problem of large-scale agents in high-dimensional environments have become a challenge. We jointly considered energy consumption, inter-group attack and defense, intra-group collision avoidance, and obstacle avoidance in their cost functions. Meanwhile, the high-dimensional state dynamics were used to describe the motion of agents under environmental interference. Then, we formulated the cooperative attack–defense evolution of large-scale agents in high-dimensional environments as a multi-population high-dimensional stochastic mean-field game (MPHD-MFG), which significantly reduced the communication frequency and computational complexity. We tractably solved the MPHD-MFG with a generative-adversarial-network (GAN)-based method using the MFGs’ underlying variational primal–dual structure. Based on our approach, we carried out an integrative experiment in which we analytically showed the fast convergence of our cooperative attack–defense evolution algorithm by the convergence of the Hamilton–Jacobi–Bellman equation’s residual errors. The experiment also showed that a large number of drones can avoid obstacles and smoothly evolve their attack and defense behaviors while minimizing their energy consumption. In addition, the comparison with the baseline methods showed that our approach is advanced.

Dual Task Framework for Improving Persona-Grounded Dialogue Dataset

This paper introduces a simple yet effective data-centric approach for the task of improving persona-conditioned dialogue agents. Prior model-centric approaches unquestioningly depend on the raw crowdsourced benchmark datasets such as Persona-Chat. In contrast, we aim to fix annotation artifacts in benchmarking, which is orthogonally applicable to any dialogue model. Specifically, we augment relevant personas to improve dialogue dataset/agent, by leveraging the primal-dual structure of the two tasks, predicting dialogue responses and personas based on each other. Experiments on Persona-Chat show that our approach outperforms pre-trained LMs by an 11.7 point gain in terms of accuracy.

Energy-Efficient Velocity Control for Massive Numbers of UAVs: A Mean Field Game Approach

In this paper, we propose an energy-efficient velocity control algorithm for a large number of UAVs based on mean field games. In particular, we first formulate the velocity control problem for a large number of UAVs as a differential game, where we jointly consider the energy consumption, channel capacity, and obstacle avoidance in the cost function. Meanwhile, the state dynamics are used to describe the motion of UAVs under the influence of the wind. Then we derive the corresponding mean field game for a large number of UAVs and solve it with the G-prox primal dual hybrid gradient (PDHG) method using its underlying variational primal dual structure. Scalability analysis shows that the computational complexity of the proposed method is unrelated to the number of UAVs. Based on the PDHG method, we conduct a comprehensive experiment where we analytically show the fast convergence of our energy-efficient velocity control algorithm by the convergence of the residual errors of the Hamilton-Jacobi-Bellman equation and the Fokker-Planck-Kolmogorov equation. The experiment also shows that a large number of UAVs can avoid obstacles and provide communication services for the search and rescue team while minimizing their energy consumption.

Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games

We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean-field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are not approachable with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle-point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.

Generalised primal-dual grids for unstructured co-volume schemes

The generation of high-quality staggered unstructured grids is considered, leading to the development of a new optimisation-based strategy designed to construct weighted ‘Regular-Power’ tessellations appropriate for co-volume type numerical discretisation techniques. This new framework aims to extend the conventional Delaunay–Voronoi primal-dual structure; seeking to assemble generalised orthogonal tessellations with enhanced geometric quality. The construction of these grids is motivated by the desire to improve the performance and accuracy of numerical methods based on unstructured co-volume type schemes, including various staggered grid techniques for the simulation of fluid dynamics and hyperbolic transport. In this study, a new hybrid optimisation strategy is proposed; seeking to optimise the geometry, topology and weights associated with general, two-dimensional Regular-Power tessellations using a combination of gradient-ascent and energy-based techniques. The performance of this new method is tested experimentally, with a range of complex, multi-resolution primal-dual grids generated for various coastal and regional ocean modelling applications.

Stabilized nonconforming finite element methods for data assimilation in incompressible flows

We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes equations. The method uses a primal dual structure that allows for th ...

Replica analysis for the duality of the portfolio optimization problem.

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we analyzed in a previous study, we herein consider a primal-dual problem in which the investment risk minimization problem with budget and expected return constraints is regarded as the primal problem, and the expected return maximization problem with budget and investment risk constraints is regarded as the dual problem. With respect to these optimal problems, we analyze a quenched disordered system involving both of these optimization problems using the approach developed in statistical mechanical informatics and confirm that both optimal portfolios can possess the primal-dual structure. Finally, the results of numerical simulations are shown to validate the effectiveness of the proposed method.

Critical solutions of nonlinear equations: stability issues

It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain so-called critical multipliers. This special subset of Lagrange multipliers defines, to a great extent, stability pattern of the solution in question subject to parametric perturbations. Criticality of a Lagrange multiplier can be equivalently characterized by the absence of the local Lipschitzian error bound in terms of the natural residual of the optimality system. In this work, taking the view of criticality as that associated to the error bound, we extend the concept to general nonlinear equations (not necessarily with primal–dual optimality structure). Among other things, we show that while singular noncritical solutions of nonlinear equations can be expected to be stable only subject to some poor “asymptotically thin” classes of perturbations, critical solutions can be stable under rich classes of perturbations. This fact is quite remarkable, considering that in the case of nonisolated solutions, critical solutions usually form a thin subset within all the solutions. We also note that the results for general equations lead to some new insights into the properties of critical Lagrange multipliers (i.e., solutions of equations with primal–dual structure).

Guaranteed upper–lower bounds on homogenized properties by FFT-based Galerkin method

Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast Fourier Transform (FFT) homogenization scheme by Vond\v{r}ejc et al. (2014), which is based on the conforming Galerkin approximation with trigonometric polynomials. Upper-lower bounds are obtained by adjusting the primal-dual finite element framework developed independently by Dvo\v{r}\'{a}k (1993) and Wieckowski (1995) to the FFT-based Galerkin setting. We show that the discretization procedure differs for odd and non-odd number of grid points. Thanks to the Helmholtz decomposition inherited from the continuous formulation, the duality structure is fully preserved for the odd discretizations. In the latter case, a more complex primal-dual structure is observed due to presence of the trigonometric polynomials associated with the Nyquist frequencies. These theoretical findings are confirmed with numerical examples. To conclude, the main advantage of the FFT-based approach over conventional finite-element schemes is that the primal and the dual problems are treated on the same basis, and this property can be extended beyond the scalar elliptic setting.

Generalized Flows for Optimal Inference in Higher Order MRF-MAP.

Use of higher order clique potentials in MRF-MAP problems has been limited primarily because of the inefficiencies of the existing algorithmic schemes. We propose a new combinatorial algorithm for computing optimal solutions to 2 label MRF-MAP problems with higher order clique potentials. The algorithm runs in time O(2(k)n(3)) in the worst case (k is size of clique and n is the number of pixels). A special gadget is introduced to model flows in a higher order clique and a technique for building a flow graph is specified. Based on the primal dual structure of the optimization problem, the notions of the capacity of an edge and a cut are generalized to define a flow problem. We show that in this flow graph, when the clique potentials are submodular, the max flow is equal to the min cut, which also is the optimal solution to the problem. We show experimentally that our algorithm provides significantly better solutions in practice and is hundreds of times faster than solution schemes like Dual Decomposition [1], TRWS [2] and Reduction [3], [4], [5]. The framework represents a significant advance in handling higher order problems making optimal inference practical for medium sized cliques.

Weighted Triangulations for Geometry Processing

In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing.

A path to the Arrow–Debreu competitive market equilibrium

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow–Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of $${O(n^{4}log(1/\epsilon}$$)) for computing an $${\epsilon}$$ -equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n4L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound $$O(n^{8}log(1/\epsilon$$)) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow–Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method.

A Class of Active-Set Newton Methods for Mixed ComplementarityProblems

Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are individually and collectively strictly weaker than the property of semistability of a solution. Thus the local superlinear convergence conditions of the presented method are weaker than conditions required for the semismooth (generalized) Newton methods applied to MCP reformulations. Moreover, they are also weaker than convergence conditions of the linearization (Josephy--Newton) method. For the special case of optimality systems with primal-dual structure, we further consider the question of superlinear convergence of primal variables. We illustrate our theoretical results with numerical experiments on some specially constructed MCPs whose solutions do not satisfy the usual regularity assumptions.

Deformation texture prediction: from the Taylor model to the advanced Lamel model

The paper reports on a recent effort to develop a statistical (or Monte-Carlo) model for quantitative deformation texture prediction which is yet fast enough for implementation in every Gauss point of an FE simulation of a metal-forming process. The principles of Taylor-type models for the prediction of deformation textures of polycrystalline materials are reminded. This includes the full-constraints Taylor theory (every grain of a polycrystal undergoes the same plastic deformation), classical Relaxed Constraints Taylor theory (one or two of the components of the local velocity gradient tensor need not be the same for all grains) and multi-grain models (LAMEL model; mentioning of GIA model). The primal–dual structure of the equations relating strain rates with slip rates, and those relating stresses and resolved shear stresses on slip systems, is made clear. It is then possible to describe the basic philosophy and the mathematical implementation of a new model, called “advanced Lamel model” (ALAMEL). This model is more generally applicable than the previously developed LAMEL model, which is only valid for rolling. Both take interactions between neighbouring grains into account. Finally, quantitative comparisons are given between experimentally observed rolling textures and the predictions of the new model, as well as of other models: full-constraints and relaxed constraints Taylor, LAMEL, GIA, visco-plastic self-consistent and crystal plasticity finite element (CPFEM) models. This was done for IF steel (one thickness reduction) and for two aluminium alloys: AA1200 (five thickness reductions) and AA5182 (one thickness reduction). It was found that for AA1200, the new model is on average the best; for the two other cases, it is among the best models, but the LAMEL or CPFEM models are better. These results suggest that in spite of all simplifications, the ALAMEL model captures (and identifies) the domination mechanisms controlling the development of deformation textures in cubic metals.