Unique Minimum Research Articles

An optimal autoconvolution inequality

AbstractLet $\mathcal {F}$ denote the set of functions $f \colon [-1/2,1/2] \to \mathbb {R}_{\geq 0}$ such that $\int f = 1$ . We determine the value of $\inf _{f \in \mathcal {F}} \| f \ast f \|_2^2$ up to a $4 \cdot 10^{-6}$ error, thereby making progress on a problem asked by Ben Green. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain improvements on the maximum size of $B_h[g]$ sets for $(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$ .

HPX filter: a hybrid of Hodrick–Prescott filter and multiple regression

Abstract This paper considers an extension of Hodrick–Prescott (HP) filter. It is a hybrid of HP filter and multiple regression. We refer to the filter as “HPX filter”. It is well known that HP filter has a unique global minimizer and the solution can be represented in matrix notation explicitly. Does HPX filter also have a unique global minimizer? Is it accomplished without any additional assumptions? Can the solution be expressed in matrix notation explicitly? In this paper, we answer these questions. In addition, this paper (i) provides an alternative perspective on the filter by representing it as a generalized ridge regression and (ii) gives an extension of it, which is a hybrid of Whittaker–Henderson method of graduation and multiple regression.

Smoothed Online Optimization with Unreliable Predictions

We consider online optimization with switching costs in a normed vector space (X, ||·||) wherein, at each time t, a decision maker observes a non-convex hitting cost function ƒ : t X →[0, ∞] and must decide upon some xt∈X→, paying ƒt (xt) + || xt-xt-1||, where ||·|| characterizes the switching cost. Throughout, we assume that ƒt is globally α-polyhedral, i.e., ƒt has a unique minimizer υt ∈X, and, for all x ∈ X, ƒ t) (x) ≥ ƒt + α · ||x - υ t. Moreover, we assume that the decision maker has access to an untrusted prediction xt of the optimal decision during each round, such as the decision suggested by a black-box AI tool.

Fast iterative method for local steric Poisson–Boltzmann theories in biomolecular solvation

This work proposes a fast iterative method for local steric Poisson–Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency are achieved by treating ionic concentrations as scalar implicit functions of the electrostatic potential, though such functions are only numerically achievable. The existence, uniqueness, boundness, and smoothness of such functions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

Semiclassical analysis of quantum asymptotic fields in the Yukawa theory

In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics, in the semiclassical regime $\hslash\to 0$, with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schr\odinger-Klein-Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schr\odinger-Klein-Gordon equation. Moreover, we prove that the asymptotic vacuum states of the Yukawa model have a phase space concentration property around classical radiationless solutions. Under further assumptions, we show that the S-KG energy admits a unique minimizer modulo symmetries and identify exactly the semiclassical measure of Yukawa ground states. Some additional consequences of asymptotic completeness are also discussed, and some further open questions are raised.

Canards in a bottleneck

In this paper, we investigate the stationary profiles of a nonlinear Fokker–Planck equation with small diffusion and nonlinear inflow and outflow boundary conditions. We consider corridors with a bottleneck whose width has a unique global nondegenerate minimum in the interior. In the small diffusion limit, the profiles are obtained constructively by using methods from geometric singular perturbation theory (GSPT). We identify three main types of profiles corresponding to: (i) high density in the domain and a boundary layer at the entrance, (ii) low density in the domain and a boundary layer at the exit, and (iii) transitions from high density to low density inside the bottleneck with boundary layers at the entrance and exit. Interestingly, solutions of the last type involve canard solutions generated at the narrowest point of the bottleneck. We obtain a detailed bifurcation diagram of these solutions in terms of the inflow and outflow rates. The analytic results based on GSPT are further corroborated by computational experiments investigating corridors with bottlenecks of variable width.

Optimal Pricing With Weighted Factors in a Product Transportation System Based on the Min-Plus Fuzzy Relation Inequality

In this work, we introduce the fuzzy relation inequalities with min-plus composition for describing the pricing problem in a product transportation system. Some basic properties of the min-plus inequalities system are investigated. We define the concept of feasible index chain and further investigate the corresponding quasi-maximal solution. Moreover, it is found that the solution set of a min-plus system could be generated by a unique minimum solution and a finite number of quasi-maximal solutions. As a consequence, one is able to find the complete solution set of a min-plus system. Besides, motivated by the optimal managerial objective, we establish an optimization model with the min-plus inequalities constraints. A so-called FIS-based algorithm is developed for searching an optimal solution of our studied optimization model. We have provided a simple numerical example for checking the effectiveness of our proposed FIS-based algorithm. The obtained optimal solution would be helpful for the system manager, as an optimal pricing scheme.

Sharp, strong and unique minimizers for low complexity robust recovery

Abstract In this paper, we show the important roles of sharp minima and strong minima for robust recovery. We also obtain several characterizations of sharp minima for convex regularized optimization problems. Our characterizations are quantitative and verifiable especially for the case of decomposable norm regularized problems including sparsity, group-sparsity and low-rank convex problems. For group-sparsity optimization problems, we show that a unique solution is a strong solution and obtains quantitative characterizations for solution uniqueness.

Local convexity of the TAP free energy and AMP convergence for Z2-synchronization

We study mean-field variational Bayesian inference using the TAP approach, for Z2-synchronization as a prototypical example of a high-dimensional Bayesian model. We show that for any signal strength λ>1 (the weak-recovery threshold), there exists a unique local minimizer of the TAP free energy functional near the mean of the Bayes posterior law. Furthermore, the TAP free energy in a local neighborhood of this minimizer is strongly convex. Consequently, a natural-gradient/mirror-descent algorithm achieves linear convergence to this minimizer from a local initialization, which may be obtained by a constant number of iterations of Approximate Message Passing (AMP). This provides a rigorous foundation for variational inference in high dimensions via minimization of the TAP free energy. We also analyze the finite-sample convergence of AMP, showing that AMP is asymptotically stable at the TAP minimizer for any λ>1, and is linearly convergent to this minimizer from a spectral initialization for sufficiently large λ. Such a guarantee is stronger than results obtainable by state evolution analyses, which only describe a fixed number of AMP iterations in the infinite-sample limit. Our proofs combine the Kac–Rice formula and Sudakov–Fernique Gaussian comparison inequality to analyze the complexity of critical points that satisfy strong convexity and stability conditions within their local neighborhoods.

Mass concentration and uniqueness of ground states for mass subcritical rotational nonlinear Schrödinger equations

This paper considers ground states of mass subcritical rotational nonlinear Schrödinger equation −Δu+V(x)u+iΩ(x⊥⋅∇u)=μu+ρp−1|u|p−1uinR2,where V(x) is an external potential, Ω>0 characterizes the rotational velocity of the trap V(x), 1<p<3, and ρ>0 describes the strength of the attractive interactions. It is shown that ground states of the above equation can be described equivalently by minimizers of the L2-constrained variational problem. We prove that minimizers exist for any ρ∈(0,∞) when 0<Ω<Ω∗, where 0<Ω∗≔Ω∗(V)<∞ denotes the critical rotational velocity. While Ω>Ω∗, there admits no minimizers for any ρ∈(0,∞). For fixed 0<Ω<Ω∗, by using energy estimates and blow-up analysis, we also analyze the mass concentration behavior of minimizers as ρ→∞. Finally, we prove that up to a constant phase, there exists a unique minimizer when Ω∈(0,Ω∗) is fixed and ρ>0 is large enough.

Generalized Gearhart-Koshy acceleration for the Kaczmarz method

The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance to the desired solution in the direction of a full Kaczmarz step. The present paper generalizes this technique to an acceleration scheme that minimizes the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently. When only a single Kaczmarz cycle is considered, the proposed affine search is more effective than the Gearhart-Koshy/Tam line-search, which in turn is more effective than the underlying Kaczmarz method. A numerical experiment from the context of computerized tomography suggests that the proposed affine search has the potential to outperform the the Gearhart-Koshy/Tam line-search and the underlying Kaczmarz method in terms of the computational cost that is needed to achieve a given error tolerance.

Well-posedness and regularity for a polyconvex energy

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the H1-projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to construct Lr-solutions of the Navier–Stokes equation (NSE) for a short time interval.

Regularization when modeling with biased simulation data as a prior

Embedding physical knowledge in system identification increases the generalization capabilities of the identified models. For complex engineering systems, such as a process plant, the most complete and detailed quantitative description of the existing physical and structural knowledge is often provided by a simulator. We describe the procedure of fusing simulated data with measurement data via L2 regularization for models that are linear in the parameters. We characterize how the MSE minimization problem in this framework is nontrivial, and show that for certain realizations of the data there is no unique minimum of the MSE w.r.t. the regularization parameter. In these cases the MSE can even increase to larger values than both the variance and the bias, which is counter-intuitive. We show how this issue appears less frequently with more data, even though multiple minima can occur for any realization of the data. However, we show also that the Stein effect is present regardless, so that it is always possible to decrease the MSE with careful selection of the regularization parameter, i.e., information fusion may always be beneficial.

Magnetic anomaly inversion through the novel barnacles mating optimization algorithm

Dealing with the ill-posed and non-unique nature of the non-linear geophysical inverse problem via local optimizers requires the use of some regularization methods, constraints, and prior information about the Earth's complex interior. Another difficulty is that the success of local search algorithms depends on a well-designed initial model located close to the parameter set providing the global minimum. On the other hand, global optimization and metaheuristic algorithms that have the ability to scan almost the entire model space do not need an assertive initial model. Thus, these approaches are increasingly incorporated into parameter estimation studies and are also gaining more popularity in the geophysical community. In this study we present the Barnacles Mating Optimizer (BMO), a recently proposed global optimizer motivated by the special mating behavior of barnacles, to interpret magnetic anomalies. This is the first example in the literature of BMO application to a geophysical inverse problem. After performing modal analyses and parameter tuning processes, BMO has been tested on simulated magnetic anomalies generated from hypothetical models and subsequently applied to three real anomalies that are chromite deposit, uranium deposit and Mesozoic dike. A second moving average (SMA) scheme to eliminate regional anomalies from observed anomalies has been examined and certified. Post-inversion uncertainty assessment analyses have been also implemented to understand the reliability of the solutions achieved. Moreover, BMO’s solutions for convergence rate, stability, robustness and accuracy have been compared with the solutions of the commonly used standard Particle Swarm Optimization (sPSO) algorithm. The results have shown that the BMO algorithm can scan the model parameter space more extensively without affecting its ability to consistently approach the unique global minimum in this presented inverse problem. We, therefore, recommend the use of competitive BMO in model parameter estimation studies performed with other geophysical methods.

An Ising machine based on networks of subharmonic electrical resonators

Combinatorial optimization problems are difficult to solve with conventional algorithms. Here we explore networks of nonlinear electronic oscillators evolving dynamically towards the solution to such problems. We show that when driven into subharmonic response, such oscillator networks can minimize the Ising Hamiltonian on non-trivial antiferromagnetically-coupled 3-regular graphs. In this context, the spin-up and spin-down states of the Ising machine are represented by the oscillators’ response at the even or odd driving cycles. Our experimental setting of driven nonlinear oscillators coupled via a programmable switch matrix leads to a unique energy minimizer when one exists, and probes frustration where appropriate. Theoretical modeling of the electronic oscillators and their couplings allows us to accurately reproduce the qualitative features of the experimental results and extends the results to larger graphs. This suggests the promise of this setup as a prototypical one for exploring the capabilities of such an unconventional computing platform.

Exploring the strong-coupling region of SU(N) Seiberg-Witten theory

We consider the Seiberg-Witten solution of pure mathcal{N} = 2 gauge theory in four dimensions, with gauge group SU(N). A simple exact series expansion for the dependence of the 2(N − 1) Seiberg-Witten periods aI(u), aDI(u) on the N − 1 Coulomb-branch moduli un is obtained around the ℤ2N-symmetric point of the Coulomb branch, where all un vanish. This generalizes earlier results for N = 2 in terms of hypergeometric functions, and for N = 3 in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the Kähler potential K = frac{1}{2}{sum}_I Im( overline{a} IaDI), which is single valued on the Coulomb branch. Evidence is presented that K is a convex function, with a unique minimum at the ℤ2N-symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing Kähler potential.

Equally spaced points are optimal for Brownian Bridge kernel interpolation

In this paper we show how ideas from spline theory can be used to construct a local basis for the space of translates of a general iterated Brownian Bridge kernel kβ,ɛ for β∈N, ɛ≥0. In the simple case β=1, we derive an explicit formula for the corresponding Lagrange basis, which allows us to solve interpolation problems without inverting any linear system.We use this basis to prove that interpolation with k1,ɛ is uniformly stable, i.e., the Lebesgue constant is bounded independently of the number and location of the interpolation points, and that equally spaced points are the unique minimizers of the associated power function, and are thus error optimal. In this derivation, we investigate the role of the shape parameter ɛ>0, and discuss its effect on these error and stability bounds.Some of the ideas discussed in this paper could be extended to more general Green kernels.

Convergence rates of Gibbs measures with degenerate minimum

We study convergence rates of Gibbs measures, with density proportional to exp(−f(x)∕t), as t→0 where f: Rd→R admits a unique global minimum at x⋆. We focus on the case where the Hessian is not definite at x⋆. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of f at x⋆, which depends on every coordinate. We give an algorithm yielding such an expansion if the polynomial order of x⋆ is no more than 8, in connection with Hilbert’s 17th problem. However, we prove that the case where the order is 10 or higher is fundamentally different and that further assumptions are needed. We then give the rate of convergence of Gibbs measures using this expansion. Finally we adapt our results to the multiple well case.

Stability of Monotone, Nonnegative, and Compactly Supported Vorticities in the Half Cylinder and Infinite Perimeter Growth for Patches

We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $\{x_{1}<\alpha\},\; \alpha>0$. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.

On local quantum Gibbs states

We address in this work the problem of minimizing quantum entropies under local constraints. We suppose that macroscopic quantities, such as the particle density, current, and kinetic energy, are fixed at each point of Rd and look for a density operator over L2(Rd), minimizing an entropy functional. Such minimizers are referred to as local Gibbs states. This setting is in contrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises, for instance, in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.