Contributions to the moment-SOS approach in global polynomial optimization

L''Optimisation Polynomiale' s'intéresse aux problèmes d'optimisation P de la forme min {f(x): x dans K} où f est un polynôme et K est un ensemble semi-algébrique de base, c'est-à-dire défini par un nombre fini de contraintes inégalité polynomiales, K={x dans Rn : gj(x) <= 0}. Cette sous discipline de l'optimisation a émergé dans la dernière décennie grâce à la combinaison de deux facteurs: l'existence de certains résultats puissants de géométrie algébrique réelle et la puissance de l'optimisation semidéfinie (qui permet d'exploiter les premiers). Il en a résulté une méthodologie générale (que nous appelons ``moments-SOS') qui permet d'approcher aussi près que l'on veut l'optimum global de P en résolvant une hiérarchie de relaxations convexes. Cependant, chaque relaxation étant un programme semi-défini dont la taille augmente avec le rang dans la hiérarchie, malheureusement, au vu de l'état de l'art actuel des progiciels de programmation semidéfinie, cette méthodologie est pour l'instant limitée à des problèmes P de taille modeste sauf si des symétries ou de la parcimonie sont présentes dans la définition de P. Cette thèse essaie donc de répondre à la question: Peux-t-on quand même utiliser la méthodologie moments-SOS pour aider à résoudre P même si on ne peut résoudre que quelques (voire une seule) relaxations de la hiérarchie? Et si oui, comment? Nous apportons deux contributions: I. Dans une première contribution nous considérons les problèmes non convexes en variables mixtes (MINLP) pour lesquelles dans les contraintes polynomiales {g(x) <=0} où le polynôme g n'est pas concave, g est concerné par peu de variables. Pour résoudre de tels problèmes (de taille est relativement importante) on utilise en général des méthodes de type ``Branch-and-Bound'. En particulier, pour des raisons d'efficacité évidentes, à chaque nœud de l'arbre de recherche on doit calculer rapidement une borne inférieure sur l'optimum global. Pour ce faire on utilise des relaxations convexes du problème obtenues grâce à l'utilisation de sous estimateurs convexes du critère f (et des polynômes g pour les contraintes g(x)<= 0 non convexes). Notre contribution est de fournir une méthodologie générale d'obtention de tels sous estimateurs polynomiaux convexes pour tout polynôme g, sur une boite. La nouveauté de notre contribution (grâce à la méthodologie moment-SOS) est de pouvoir minimiser directement le critère d'erreur naturel qui mesure la norme L_1 de la différence f-f' entre f et son sous estimateur convexe polynomial f'. Les résultats expérimentaux confirment que le sous estimateur convexe polynomial que nous obtenons est nettement meilleur que ceux obtenus par des méthodes classiques de type ``alpha-BB' et leurs variantes, tant du point de vue du critère L_1 que du point de vue de la qualité des bornes inférieures obtenus quand on minimise f' (au lieu de f) sur la boite. II: Dans une deuxième contribution on considère des problèmes P pour lesquels seules quelques relaxations de la hiérarchie moments-SOS peuvent être implantées, par exemple celle de rang k dans la hiérarchie, et on utilise la solution de cette relaxation pour construire une solution admissible de P. Cette idée a déjà été exploitée pour certains problèmes combinatoire en variables 0/1, parfois avec des garanties de performance remarquables (par exemple pour le problème MAXCUT). Nous utilisons des résultats récents de l'approche moment-SOS en programmation polynomiale paramétrique pour définir un algorithme qui calcule une solution admissible pour P à partir d'une modification mineure de la relaxation convexe d'ordre k. L'idée de base est de considérer la variable x_1 comme un paramètre dans un intervalle Y_1 de R et on approxime la fonction ``valeur optimale' J(y) du problème d'optimisation paramétrique P(y)= min {f(x): x dans K; x_1=y} par un polynôme univarié de degré d fixé. Cette étape se ramène à la résolution d'un problème d'optimisation convexe (programme semidéfini). On calcule un minimiseur global y de J sur l'intervalle Y (un problème d'optimisation convexe ``facile') et on fixe la variable x_1=y. On itère ensuite sur les variables restantes x_2,...,x_n en prenant x_2 comme paramètre dans un intervalle Y_2, etc. jusqu'à obtenir une solution complète x de R^n qui est faisable si K est convexe ou dans certains problèmes en variables 0/1 où la faisabilité est facile à vérifier (e.g., MAXCUT, k-CLUSTTER, Knapsack). Sinon on utilise le point obtenu x comme initialisation dans un procédure d'optimisation locale pour obtenir une solution admissible. Les résultats expérimentaux obtenus sur de nombreux exemples sont très encourageants et prometteurs.Polynomial Optimization is concerned with optimization problems of the form (P) : f* = { f(x) with x in set K}, where K is a basic semi-algebraic set in Rn defined by K={x in Rn such as gj(x) less or equal 0}; and f is a real polynomial of n variables x = (x1, x2, ..., xn). In this thesis we are interested in problems (P) where symmetries and/or structured sparsity are not easy to detect or to exploit, and where only a few (or even no) semidefinite relaxations of the moment-SOS approach can be implemented. And the issue we investigate is: How can the moment-SOS methodology be still used to help solve such problem (P)? We provide two applications of the moment-SOS approach to help solve (P) in two different contexts. * In a first contribution we consider MINLP problems on a box B = [xL, xU] of Rn and propose a moment-SOS approach to construct polynomial convex underestimators for the objective function f (if non convex) and for -gj if in the constraint gj(x) less or equal 0, the polynomial gj is not concave. We work in the context where one wishes to find a convex underestimator of a non-convex polynomial f of a few variables on a box B of Rn. The novelty with previous works on this topic is that we want to compute a polynomial convex underestimator p of f that minimizes the important tightness criterion which is the L1 norm of (f-h) on B, over all convex polynomials h of degree d _fixed. Indeed in previous works for computing a convex underestimator L of f, this tightness criterion is not taken into account directly. It turns out that the moment-SOS approach is well suited to compute a polynomial convex underestimator p that minimizes the tightness criterion and numerical experiments on a sample of non-trivial examples show that p outperforms L not only with respect to the tightness score but also in terms of the resulting lower bounds obtained by minimizing respectively p and L on B. Similar improvements also occur when we use the moment-SOS underestimator instead of the aBB-one in refinements of the aBB method. * In a second contribution we propose an algorithm that also uses an optimal solution of a semidefinite relaxation in the moment-SOS hierarchy (in fact a slight modification) to provide a feasible solution for the initial optimization problem but with no rounding procedure. In the present context, we treat the first variable x1 of x = (x1, x2, ...., xn) as a parameter in some bounded interval Y of R. Notice that f*=min { J(y) : y in Y} where J is the function J(y) := inf {f(x) : x in K ; x1=y}. That is one has reduced the original n-dimensional optimization problem (P) to an equivalent one-dimensional optimization problem on an interval. But of course determining the optimal value function J is even more complicated than (P) as one has to determine a function (instead of a point in Rn), an infinite-dimensional problem. But the idea is to approximate J(y) on Y by a univariate polynomial p(y) with the degree d and fortunately, computing such a univariate polynomial is possible via solving a semidefinite relaxation associated with the parameter optimization problem. The degree d of p(y) is related to the size of this semidefinite relaxation. The higher the degree d is, the better is the approximation of J(y) by p(y) and in fact, one may show that p(y) converges to J(y) in a strong sense on Y as d increases. But of course the resulting semidefinite relaxation becomes harder (or impossible) to solve as d increases and so in practice d is fixed to a small value. Once the univariate polynomial p(y) has been determined, one computes x1* in Y that minimizes p(y) on Y, a convex optimization problem that can be solved efficiently. The process is iterated to compute x2 in a similar manner, and so on, until a point x in Rn has been computed. Finally, as x* is not feasible in general, we then use x* as a starting point for a local optimization procedure to find a final feasible point x in K. When K is convex, the following variant is implemented. After having computed x1* as indicated, x2* is computed with x1 fixed at the value x1*, and x3 is computed with x1 and x2 fixed at the values x1* and x2* respectively, etc., so that the resulting point x* is feasible, i.e., x* in K. The same variant applies for 0/1 programs for which feasibility is easy to detect like e.g., for MAXCUT, k-CLUSTER or 0/1-KNAPSACK problems

Performance of export-oriented small and medium-sized manufacturing enterprises in Viet Nam

The study recommends the formulation of policies that support the development of business linkages and networking, and which promote subcontracting arrangements between small and large enterprises or between domestic firms and foreign investment enterprises. It is also necessary to support and facilitate the direct involvement of SMMEs in exporting or indirectly through large manufacturing enterprises.Export-oriented, SME,SMME, Viet Nam

Synthesis and characterization of curcumin-phosphatidylcholine complex via solution enhanced dispersion by supercritical CO2

In order to enhance the bioavailability of poorly water-soluble curcumin, solution enhanced dispersion by supercritical carbon dioxide (CO2) (SEDS) was employed to prepare curcumin-phosphatidyl choline complex. The reaction parameters were varied and investigated. The typical parameters were determined as follow:  P = 200 bar, T = 60 °C, gas flow rate 10 mL.min-1, molar ratio of phosphatidylcholine:curcumin 2:1, ratio of dichloromethane/scCO2 15 % (w/w). The characteristics of complexation product were measured such as the water solubility, natural stability, UV-Vis, FTIR and DSC. The FE-SEM image shows that curcumin-phosphatidylcholine complex appeared in two types of spherical shape with the big particle size of  complex approximately 0.9 µm and nano curcumin with particles size of approximately 100 nm. This study revealed that supercritical CO2 technologies had a great potential in fabricating complex and improving the bioavailability of poorly water-soluble drugs. Keywords. Curcumin, water soluble curcumin, phosphatidylcholine, CO2 supercritical fluids

Corona based air-flow using parallel discharge electrodes

A novel air-flow generator based on the effect of ion wind has been developed by the simultaneous generation of both positive and negative ions using two electrodes of opposite polarity placed in parallel. Unlike the conventional unipolar-generators, this bipolar configuration creates an ion wind, which moves away from both electrodes and yields a very low net charge on the device. The electro-hydrodynamic behavior of air-flow has been experimentally and numerically studied. The velocity of ion wind reaches values up to 1.25 m/s using low discharge current 5 mu-A with the kinetic conversion efficiency of 0.65% and the released net charge of �30 fA, 8 orders of magnitude smaller compared with the discharge current. Due to easy scalability and low net charge, the present configuration is beneficial to applications with space constraints and/or where neutralized discharge process is required, such as inertial fluidic units, circulatory flow heat transfer, electrospun polymer nanofiber to overcome the intrinsically instability of the process, or the formation of low charged aerosol

A closed device to generate vortex flow using PZT

This paper reports for the first time a millimeter scale fully packaged device which generates a vortex flow of high velocity. The flow which is simply actuated by a PZT diaphragm circulates with a higher velocity after each actuating circle to form a vortex in a desired chamber. The design of such device is firstly conducted by a numerical analysis using OpenFOAM. Several numerical results are considered as the base of our experiment where a flow vortex is observed by a high speed camera. The present device is potential in various applications related to the inertial sensing, fluidic amplifier and micro/nano particle trapping and mixing

Tri-axis convective accelerometer with closed-loop heat source

In this paper, we report the details and findings of a study on tri-axis convective accelerometer, which is designed with the closed-loop type heat source and thermal sensing hotwire elements. The closed-loopheat source enhances the convective flow to the central part where a hotwire is placed to measure the vertical component of acceleration. The simulation was conducted using numerical analysis, and the devicewas prototyped by additive manufacturing. The device, functioning as a tilt sensor and an accelerometer,was tested up to acceleration of 20 g. The experiments were successfully conducted and the experimental results agreed reasonably with those obtained by numerical analysis. The results demonstrated that the closed-loop heat source could reduce the cross effect between the acceleration components. The scalefactor and cross-sensitivity had the values of 0.26 micro�V/g and 1.2%, respectively. The cross-sensitivity andthe effects of heating power were also investigated in this study

A circulatory ionic wind for inertial sensing application

A novel gyroscope using circulatory electro-hydrodynamics flow in a confined space is presented for the first time. The configuration of the new gyroscope includes three point-ring corona discharge actuators that generate ion flows in three separated sub-channels. The three ion flows then merge together when going through a nozzle of the main chamber entrance and create a jet flow. In the new configuration, the residual charge of ion wind flow is removed by a master-ring electrode located at one end of the main chamber. Under the effect of the angular speed of gyroscope, the jet flow is deflected and this deflection is sensed using hotwires. The results, which are consistently acquired by both the numerical simulation and experiment on our prototypes, demonstrate the repeatability and stability of the new approach. Since the ion wind can be generated by a minimum power, the present configuration-based device does not require any vibrating component. Thus, the device is robust, cost, and energy-effective

The Investigation on the Fabrication and Characterization of the Multicomponent Ceramics Based on PZT and the Relaxor PZN-PMnN Ferroelectric Materials

This chapter presents the investigation of fabrication and the physical properties of the Pb(Zr1−xTix)O3-Pb(Zn1/3Nb2/3)O3-Pb(Mn1/3Nb2/3)O3 multicomponent ceramics. The multicomponent yPb(Zr1−xTix)O3-(0.925 − y)Pb(Zn1/3Nb2/3)O3-0.075Pb(Mn1/3Nb2/3)O3 (PZT-PZN-PMnN) ceramics were synthesized by conventional solid-state reaction method (MO) combined with the B-site oxide mixing technique (BO). Research results show that the electrical properties of PZT-PZN-PMnN ceramics are optimal at a PZT content of 0.8 mol and Zr/Ti ratio of 48/52. At these contents, the ceramics have the following optimal properties: electromechanical coupling factor, kp = 0.62 and kt = 0.51; piezoelectric constant (d31) of 130 pC/N; mechanical quality factor (Qm) of 1112; dielectric loss (tan δ) of 0.005; high remanent polarization (Pr) of 30.4 μC.cm−2; and low coercive field (EC) of 6.2 kV.cm−1. Investigation of the domain structure of the two ferroelectric phases (tetragonal and rhombohedral) in the ZnO-doped PZT-PZN-PMnN with compositions at near the morphotropic phase boundary is described as follows: the 90 and 180° domains exist in the tetragonal phase, while the 71, 109, and 90° domains are located in the rhombohedral phase, and the widths of these domains were about 100 nm. Besides, the ceramics exhibited excellent temperature stability, which makes them a promising material for high-intensity ultrasound applications

Flat Seeking Bayesian Neural Networks

Bayesian Neural Networks (BNNs) provide a probabilistic interpretation for deep learning models by imposing a prior distribution over model parameters and inferring a posterior distribution based on observed data. The model sampled from the posterior distribution can be used for providing ensemble predictions and quantifying prediction uncertainty. It is well-known that deep learning models with lower sharpness have better generalization ability. However, existing posterior inferences are not aware of sharpness/flatness in terms of formulation, possibly leading to high sharpness for the models sampled from them. In this paper, we develop theories, the Bayesian setting, and the variational inference approach for the sharpness-aware posterior. Specifically, the models sampled from our sharpness-aware posterior, and the optimal approximate posterior estimating this sharpness-aware posterior, have better flatness, hence possibly possessing higher generalization ability. We conduct experiments by leveraging the sharpness-aware posterior with state-of-the-art Bayesian Neural Networks, showing that the flat-seeking counterparts outperform their baselines in all metrics of interest.Comment: Accepted at NeurIPS 202