Second-order Conditions Research Articles

Second-order optimality conditions for singular extremals in optimal control problems with equality endpoint constraints

In this paper we use the method of finite-dimensional approximations to derive the second-order necessary optimality conditions for singular Pontryagin local minimizers in optimal control problems with equality endpoint constraints.

On the No-Gap Second-Order Optimality Conditions for a Discrete Optimal Control Problem with Mixed Constraints

Motivated by our recent works on optimality conditions in discrete optimal control problems under a nonconvex cost function, in this paper, we study second-order necessary and sufficient optimality conditions for a discrete optimal control problem with a nonconvex cost function and state-control constraints. By establishing an abstract result on second-order optimality conditions for a mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain "no-gap" between second-order optimality conditions.

Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems

Using an augmented Lagrangian approach, we study the existence of augmented Lagrange multipliers of a semi-infinite programming problem and discuss their characterizations in terms of saddle points. In the case of a sharp Lagrangian, we obtain a first-order necessary condition for the existence of an augmented Lagrange multiplier for the semi-infinite programming problem and some first-order sufficient conditions by assuming inf-compactness of the data functions and the extended Mangasarian---Fromovitz constraint qualification. Using a valley at 0 augmenting function and assuming suitable second-order sufficient conditions, we obtain the existence of an augmented Lagrange multiplier for the semi-infinite programming problem.

Experimental and analytical investigation of adsorption effects on shale gas transport in organic nanopores

Gas production from shale gas reservoirs is determined by original gas-in-place and gas transport mechanism in shale. Gas storage in shale consists of three different states: first, free gas in pores and natural fractures; second, adsorbed gas on the organic and inorganic pore walls; and third, absorbed or dissolved gas into formation water and organic matter. On the other hand, gas transport in shale may be dominated by noncontinuum flow in nanopores, which depends on gas slippage, bulk diffusion, Knudsen diffusion and adsorbed surface diffusion. In this study, a gas velocity model involving all of these parameters is proposed to describe the gas adsorption and Non-Darcy flow effects on gas transport behavior in shale gas reservoirs. Adsorption capacity of six shale core samples is measured by the gravimetric method using magnetic suspension sorption system. Regression analysis of the measured data has been performed using the Simplified Local-Density model coupled with modified Peng-Robinson Equation of State (SLD-PR). The presence of the moisture content is taken into consideration to distinguish between gas adsorption on the surface of organic matter and clay minerals. The hydrophilic feature of clay minerals and hydrophobic feature of organic matter result in water molecules preferring adsorbing on the surface of clays, while gas molecules preferring adsorbing on the surface of organic matter. In the developed velocity model, adsorbed gas is treated as mobile gas molecules adsorbing on the surface of organic nanopores, which moves by means of adsorbed surface diffusion. The surface diffusion velocity is characterized by adsorbed surface diffusion coefficient. Simultaneously, bulk gas is governed by second-order gas slippage, which is calculated using Navier-Stokes (N-S) equation associated with second-order slip boundary condition. Finally, the adsorbed surface diffusion is incorporated into bulk gas flow using Langmuir slip model to account for the total gas flow in the organic nanopores. The results of this study indicate that adsorbed surface diffusion contributes to the enhancement or impediment of total gas flow velocity in organic nanopores. Therefore, properly handling the adsorption effect on gas transport in organic nanopores is critical to correctly understand gas production from shale gas reservoirs.

Optimality conditions for sparse nonlinear programming

The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable function subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Fr${\rm\acute{e}}$chet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn-Tucker (KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

Pointwise second-order necessary optimality conditions and second-order sensitivity relations in optimal control

This paper is devoted to pointwise second-order necessary optimality conditions for the Mayer problem arising in optimal control theory. We first show that with every optimal trajectory it is possible to associate a solution p(⋅) of the adjoint system (as in the Pontryagin maximum principle) and a matrix solution W(⋅) of an adjoint matrix differential equation that satisfy a second-order transversality condition and a second-order maximality condition. These conditions seem to be a natural second-order extension of the maximum principle. We then prove a Jacobson like necessary optimality condition for general control systems and measurable optimal controls that may be only “partially singular” and may take values on the boundary of control constraints. Finally we investigate the second-order sensitivity relations along optimal trajectories involving both p(⋅) and W(⋅).

Folded concave penalized sparse linear regression: sparsity, statistical performance, and algorithmic theory for local solutions.

This paper concerns the folded concave penalized sparse linear regression (FCPSLR), a class of popular sparse recovery methods. Although FCPSLR yields desirable recovery performance when solved globally, computing a global solution is NP-complete. Despite some existing statistical performance analyses on local minimizers or on specific FCPSLR-based learning algorithms, it still remains open questions whether local solutions that are known to admit fully polynomial-time approximation schemes (FPTAS) may already be sufficient to ensure the statistical performance, and whether that statistical performance can be non-contingent on the specific designs of computing procedures. To address the questions, this paper presents the following threefold results: (i) Any local solution (stationary point) is a sparse estimator, under some conditions on the parameters of the folded concave penalties. (ii) Perhaps more importantly, any local solution satisfying a significant subspace second-order necessary condition (S3ONC), which is weaker than the second-order KKT condition, yields a bounded error in approximating the true parameter with high probability. In addition, if the minimal signal strength is sufficient, the S3ONC solution likely recovers the oracle solution. This result also explicates that the goal of improving the statistical performance is consistent with the optimization criteria of minimizing the suboptimality gap in solving the non-convex programming formulation of FCPSLR. (iii) We apply (ii) to the special case of FCPSLR with minimax concave penalty (MCP) and show that under the restricted eigenvalue condition, any S3ONC solution with a better objective value than the Lasso solution entails the strong oracle property. In addition, such a solution generates a model error (ME) comparable to the optimal but exponential-time sparse estimator given a sufficient sample size, while the worst-case ME is comparable to the Lasso in general. Furthermore, to guarantee the S3ONC admits FPTAS.

Parabolic Second-Order Directional Differentiability in the Hadamard Sense of the Vector-Valued Functions Associated with Circular Cones

In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone. The vector-valued function comes from applying a given real-valued function to the spectral decomposition associated with circular cone. In particular, we present the exact formula of second-order tangent set of circular cone by using the parabolic second-order directional derivative of projection operator. In addition, we also deal with the relationship of second-order differentiability between the vector-valued function and the given real-valued function. The results in this paper build fundamental bricks to the characterizations of second-order necessary and sufficient conditions for circular cone optimization problems.

The Importance of Being Normal, Regular and Proper in the Calculus of Variations

Some classical references in the literature include second-order necessary conditions for the isoperimetric problem of Lagrange in the calculus of variations involving inequality and equality constraints. Those conditions take into account, in the sets of critical directions, the sign of the Lagrange multipliers of the extremal under consideration. However, they hold under a strong assumption of normality relative to a set defined only by equality constraints for active indices. In this paper, we show how the same conditions can be preserved under a weaker assumption, thus providing a wider range of applicability.

Higher-order parameter-free sufficient optimality conditions in discrete minmax fractional programming

The purpose of this paper is to establish a fairly large number of sets of second-order parameter-free sufficient optimality conditions for a discrete minmax fractional programming problem. Our effort to accomplish this goal is by utilizing various new classes of generalized second-order $(\phi,\eta,\rho,\theta,m)$-invex functions, which generalize most of the concepts available in the literature.

A second-order sequential optimality condition associated to the convergence of optimization algorithms

Sequential optimality conditions have recently played an important role on the analysis of the global convergence of optimization algorithms towards first-order stationary points, justifying their stopping criteria. In this paper we introduce a sequential optimality condition that takes into account second-order information and that allows us to improve the global convergence assumptions of several second-order algorithms, which is our main goal. We also present a companion constraint qualification that is less stringent than previous assumptions associated to the convergence of second-order methods, like the joint condition Mangasarian-Fromovitz and Weak Constant Rank. Our condition is also weaker than the Constant Rank Constraint Qualification, which associated it to the convergence of second-order algorithms. This means that we can prove second-order global convergence of well stablished algorithms even when the set of Lagrange multipliers is unbounded, which overcomes a limitation of previous results based on MFCQ. We prove global convergence of well known variations of the augmented Lagrangian and Regularized SQP methods to second-order stationary points under this new weak constraint qualification.

Second-order lower radial tangent derivatives and applications to set-valued optimization

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.

Sufficient Second-Order Conditions for Bang-Bang Control Problems

We provide sufficient optimality conditions for optimal control problems with bang-bang controls. Building on a structural assumption on the adjoint state, we additionally need a weak second-order condition. This second-order condition is formulated with functions from an extended critical cone, and it is equivalent to a formulation posed on measures supported on the set where the adjoint state vanishes. If our sufficient optimality condition is satisfied, we obtain a local quadratic growth condition in $L^1(\Omega)$.

A New Type of Identification Problems: Optimizing the Fractional Order in a Nonlocal Evolution Equation

In this paper, we consider a rather general linear evolution equation of fractional type, namely, a diffusion type problem in which the diffusion operator is the $s$th power of a positive definite operator having a discrete spectrum in ${\mathbb R}^+$. We prove existence, uniqueness, and differentiability properties with respect to the fractional parameter $s$. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the “control parameter” that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new class of identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the ...

Pointwise Second-Order Necessary Conditions for Stochastic Optimal Controls, Part II: The General Case

This paper is the second part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations (which are quite different from the case of convex control constraint), we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of the Pontryagin-type maximum principle.

Optimality conditions of second-order radial epiderivatives

In this paper, we introduce the concepts of second-order radial epiderivative and second-order generalized radial epiderivative for nonconvex set-valued maps. We give existence theorems for the second-order generalized radial epiderivatives. We also establish the second-order optimality conditions by using second-order radial epiderivatives. https://doi.org/10.28919/jmcs/3440

Numerical simulation of heat transfer in a micro-cavity

The behaviour of rarefied monatomic gas of Maxwell particles within a rectangular enclosure is investigated, with the Navier–Stokes and Fourier field of equations with first- (NSF) and second-order boundary conditions (NSF2) of the velocity slip and temperature jump, and the regularized 13 moments approach (R13). The enclosure considered has a heated bottom with lateral walls that have specular reflection. The effect of the three dimensionless parameters characterizing the simulated problem, the cavity aspect ratio, the Knudsen number, and the temperature ratio of the hot over the cold walls, on the flow and bulk quantities is examined. For the small Knudsen numbers the flow presents one type of streamlines from the cold to hot plate in both NSF and R13 theories, while by increasing the Knudsen number the flow becomes more complex and presents hot to cold flow streamlines in the extended approach of R13. These rarefaction effects cannot be predicted by the classical continuum approach of NSF. The increase of the temperature ratio in R13 affects the hot to cold flow, which begins to vanish, while this type of streamline does not appear by decreasing the aspect ratio.

On long-range dependence of random measures

AbstractThis paper deals with long-range dependence of random measures on ℝd. By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived.

Heat transfer of nanofluids considering nanoparticle migration and second-order slip velocity

The heat transfer of a magnetohydrodynamics nanofluid inside an annu-lus considering the second-order slip condition and nanoparticle migration is theoret-ically investigated. A second-order slip condition, which appropriately represents the non-equilibrium region near the interface, is prescribed rather than the no-slip condi-tion and the linear Navier slip condition. To impose different temperature gradients, the outer wall is subjected to q2, the inner wall is subjected to q1, and q1 > q2. A modified two-component four-equation non-homogeneous equilibrium model is employed for the nanofluid, which have been reduced to two-point ordinary boundary value differential equations in the consideration of the thermally and hydrodynamically fully developed flow. The homotopy analysis method (HAM) is employed to solve the equations, and the h-curves are plotted to verify the accuracy and efficiency of the solutions. Moreover, the effects of the physical factors on the flow and heat transfer are discussed in detail, and the semi-analytical relation between NuB and NBT is obtained.

Necessary second-order conditions for a weak local minimum in a problem with endpoint and control constraints

One of the first steps towards necessary second-order optimality conditions in problems with constraints was taken by Dubovitskii and Milyutin in 1965. They offered a scheme that was very effective in smooth optimization problems, but seemed to be not suitable for applications in problems with pointwise control constraints. In this article we consider a modification of the Dubovitskii–Milyutin scheme, which allows to derive necessary second-order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control.