Saddle Point Condition Research Articles

Nucleation of crystal surfaces with corner energy regularization

The thermodynamics of strongly anisotropic crystalline surfaces is analogous to that of a binary mixture exhibiting phase separation. On a metastable planar surface, formation of stable orientations requires a nucleation process, in which the energy associated with the presence of corners must be considered. In this context, a nucleation event corresponds to the formation of a critical shape for the crystalline surface before the system enters the growth regime. We first derive the Euler-Lagrange equation for crystal surface nucleation, in two dimensions, and show that the saddle-point condition corresponds to a vanishing chemical potential along this critical surface. We then perform numerical simulation of the equation of motion for the crystal surface and show that, as compared with saddle point nucleation, ridge crossing is dynamically favoured.

Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers

In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.

Unintegrated dipole gluon distribution at small transverse momentum

We derive analytical results for unintegrated color dipole gluon distribution function at small transverse momentum. By Fourier transforming the $S$-matrix for large dipoles we derive the results in the form of a series of Bells polynomials. Interestingly, when resumming the series in leading log accuracy, the results showing up striking similarity with the Sudakov form factor with role play of coupling is being done by a constant that stems from the saddle point condition along the saturation line.

Relaxation and optimization for linear-growth convex integral functionals under PDE constraints

We give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the formF[u]=∫Ωf(x,u(x))dx,u:Ω⊂Rd→RN, where f:Ω×RN→R is a convex integrand and u is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures M(Ω;RN), and the introduction of a set-valued pairing on M(Ω;RN)×L∞(Ω;RN). By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD and divergence-free spaces.

Characterization of duality for a generalized quasi-equilibrium problem

In this paper, the duality theory of a generalized quasi-equilibrium problem (also called generalized Ky Fan quasi-inequality) is investigated by using the image space approach. Generalized quasi-equilibrium problem is transformed into a minimization problem. The minimization problem is further reformulated as an image problem by virtue of linear/nonlinear separation function. The dual problem of the image problem is constructed in the image space, then zero duality gap between the image problem and its dual problem is derived under saddle point condition as well as the equivalent regular linear/nonlinear separation condition. Finally, some more sufficient conditions guaranteeing zero duality gap are also proposed.

Saddle point criteria in semi-infinite minimax fractional programming under (Φ,ρ)-invexity

Semi-infinite minimax fractional programming problems with both inequality and equality constraints are considered. The sets of parametric saddle point conditions are established for a new class of nonconvex differentiable semi-infinite minimax fractional programming problems under(?,?)-invexity assumptions. With the reference to the said concept of generalized convexity, we extend some results of saddle point criteria for a larger class of nonconvex semi-infinite minimax fractional programming problems in comparison to those ones previously established in the literature.

A mixed L2/Lα differential game approach to pursuit-evasion guidance

A differential game–based strategy, L2/Lα guidance law, is derived for a missile with large lateral acceleration capability intercepting an evading target that has limited lateral acceleration capability. The engagement is formulated as a two-person zero-sum pursuit-evasion game with a linear quadratic cost, where only the maneuverability of the evader is assumed bounded. The open-loop solution is derived via direct derivation of the lower and upper values of the game and the saddle point condition. It is shown that the existence of an open-loop saddle point solution depends on the initial conditions of the engagement. A closed-form guidance law is formulated, consisting of the optimal saddle point strategies and three proposed variants when no saddle point solution exists. Linear and nonlinear simulations are performed for the case of a pursuer with first-order control dynamics and an evader with zero-lag dynamics in order to illustrate the advantages and performance of the proposed guidance algorithm in comparison with classical optimal and differential game–based guidance laws.

Nonlinear separation in the image space with applications to constrained optimization

In this paper, by means of the image space analysis, we obtain optimality conditions for vector optimization of objective multifunction with multivalued constraints based on disjunction of two suitable subsets of the image space. By the oriented distance function a nonlinear regular separation is introduced and some optimality conditions for the constrained extremum problem are obtained. It is shown that the existence of a nonlinear separation is equivalent to a saddle point condition for the generalized Lagrangian function.

Going beyond the BCS level in the superfluid path integral: A consistent treatment of electrodynamics and thermodynamics

In this paper we show how to redress a shortcoming of the path integral scheme for fermionic superfluids and superconductors. This approach is built around a simultaneous calculation of electrodynamics and thermodynamics. An important sum rule, the compressibility sum rule, fails to be satisfied in the usual calculation of the electromagnetic and thermodynamic response at the Gaussian fluctuation level. Here we present a path integral scheme to address this inconsistency. Specifically, at the leading order we argue that the superconducting gap should be calculated using a different saddle point condition modified by the presence of an external vector potential. This leads to the well known gauge-invariant BCS electrodynamic response and is associated with the usual (mean field) expression for thermodynamics. In this way the compressibility sum rule is satisfied at the BCS level. Moreover, this scheme can be readily extended to address arbitrary higher order fluctuation theories. At any level this approach will lead to a gauge invariant and compressibility sum rule consistent treatment of electrodynamics and thermodynamics.

An Exact $$l_1$$ l 1 Penalty Approach for Interval-Valued Programming Problem

The objective of this paper is to propose an exact \(l_1\) penalty method for constrained interval-valued programming problems which transform the constrained problem into an unconstrained interval-valued penalized optimization problem. Under some suitable conditions, we establish the equivalence between an optimal solution of interval-valued primal and penalized optimization problem. Moreover, saddle-point type optimality conditions are also established in order to find the relation between an optimal solution of penalized optimization problem and saddle-point of Lagrangian function. Numerical examples are given to illustrate the derived results.

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.

Lagrange optimality system for a class of nonsmooth convex optimization

In this paper, we revisit the augmented Lagrangian method for a class of nonsmooth convex optimization. We present the Lagrange optimality system of the augmented Lagrangian associated with the problems, and establish its connections with the standard optimality condition and the saddle point condition of the augmented Lagrangian, which provides a powerful tool for developing numerical algorithms: we derive a Lagrange–Newton algorithm for the nonsmooth convex optimization, and establish the nonsingularity of the Newton system and the local convergence of the algorithm.

The Secrecy Capacity of Compound Gaussian MIMO Wiretap Channels

Strong secrecy capacity of compound wiretap channels is studied. The known lower bounds for the secrecy capacity of compound finite-state memoryless channels under discrete alphabets are extended to arbitrary uncertainty sets and continuous alphabets under the strong secrecy criterion. The conditions under which these bounds are tight are given. Under the saddle-point condition, the compound secrecy capacity is shown to be equal to that of the worst-case channel. Based on this, the compound Gaussian multiple-input multiple-output wiretap channel is studied under the spectral norm constraint and without the degradedness assumption. First, it is assumed that only the eavesdropper channel is unknown, but is known to have a bounded spectral norm (maximum channel gain). The compound secrecy capacity is established in a closed form and the optimal signaling is identified. The compound capacity equals the worst-case channel capacity and thus establishing the saddle-point property; the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it. More general uncertainty sets are considered and the existence of a maximum element is shown to be sufficient for a saddle-point to exist, so that signaling on the worst-case channel achieves the compound capacity of the whole class of channels. The case of rank-constrained eavesdropper is considered and the respective compound secrecy capacity is established. Subsequently, the case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel, is studied. Its compound secrecy capacity and the optimal signaling are established in a closed form as well, revealing the same saddle-point property. When a saddle-point exists under strong secrecy, strong and weak secrecy compound capacities are equal.

Asymptotic optimality conditions for linear semi-infinite programming

In this paper, the classical KKT, complementarity and Lagrangian saddle-point conditions are generalized to obtain equivalent conditions characterizing the optimality of a feasible solution to a general linear semi-infinite programming problem without constraint qualifications. The method of this paper differs from the usual convex analysis methods and its main idea is rooted in some fundamental properties of linear programming.

A new approach to constrained optimization via image space analysis

In this article, by introducing a class of nonlinear separation functions, the image space analysis is employed to investigate a class of constrained optimization problems. Furthermore, the equivalence between the existence of nonlinear separation function and a saddle point condition for a generalized Lagrangian function associated with the given problem is proved.

Gradient-based estimation of uncertain parameters for elliptic partial differential equations

This paper discusses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We treat the parameter identification problem as a variational problem over the appropriate stochastic Sobolev spaces and show that minimizers exist and satisfy a saddle point condition. Although a lack of regularity precludes the direct use of gradient-based optimization techniques, a spectral approximation of the observation field allows us to estimate the original problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called finite noise problem, which lends itself readily to more traditional optimization approaches. We prove that the finite noise minimizers converge to the appropriate infinite dimensional ones, and devise and analyze a stochastic augmented Lagrangian method for locating these numerically. We also discuss the numerical discretization of the finite noise problem, using sparse grid hierarchical finite elements, and present three numerical examples to illustrate our method.

Vector Variational-Like Inequalities with Constraints: Separation and Alternative

Based on the oriented distance function, a linear weak separation function and three nonlinear regular weak separation functions are introduced in reflexive Banach spaces. Particularly, a nonlinear regular weak separation function does not involve any parameters. Moreover, theorems of the weak alternative for vector variational-like inequalities with constraints are derived by the separation functions without any convexity. Saddle-point conditions, which show the equivalence between the existence of a saddle point and a (linear) nonlinear separation of two suitable subsets of the image space, are established for the linear and nonlinear regular weak separation functions, respectively. Necessary and sufficient optimality conditions for vector variational-like inequalities with constraints are also obtained via the saddle-point conditions. Finally, two gap functions for vector variational-like inequalities with constraints and their continuity are derived by using the image space analysis.

Eigenspace Stability Conditions in Mean-Field Replica Theories

I consider the eigenspace form of the saddle point conditions that apply for mean-field replica models that are characterized by a single overlap matrix order parameter. These allow for a characterization of the full set of saddle point solutions. Further, they motivate consideration of a particular class of solutions, which I call the principally-commuting set. These are the solutions that simultaneously satisfy the replica-space rotational stability constraints for all physical models. I show that this set satisfies the replica equivalence symmetry, as well as other more stringent conditions, and that it also contains the set of Parisi-type ultrametric solutions as a subset.

Point-source radiation in attenuative anisotropic media

Important insights into point-source radiation in attenuative anisotropic media can be gained by applying asymptotic methods. Analytic description of point-source radiation can help in the interpretation of the amplitude-variation-with-offset response and physical-modeling data. We derive the asymptotic Green’s function in homogeneous, attenuative, arbitrarily anisotropic media using the steepest-descent method. The saddle-point condition helps describe the behavior of the slowness and group-velocity vectors of the far-field P-wave. Numerical results from the asymptotic analysis compare well with those obtained by the ray-perturbation method for P-waves in transversely isotropic media.

On Saddle Points in Semidefinite Optimization via Separation Scheme

This paper aims at investigating saddle point conditions for augmented Lagrangian functions for semidefinite optimization problems. By means of the image space analysis, the existence of a saddle point is shown to be equivalent to a regular weak nonlinear separation of two suitable subsets in the image space (IS) associated with the given problem. Especially, three classes of augmented Lagrangians based on smooth spectral penalty functions can be derived, as particular cases, from a nonlinear separation scheme in the IS. Without requiring the strict complementarity, it is proved that, under strong second-order sufficiency conditions, all these augmented Lagrangian functions admit a local saddle point, and their Hessians become positive definite in a neighborhood of a local optimal point of the original problem. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set.