Second-order Optimality Conditions Research Articles

Using negative curvature in solving nonlinear programs

Minimization methods that search along a curvilinear path composed of a non-ascent negative curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent applications, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions.

The Foundations of J M Keynes's IS-LM Model in Chapter 15 of the General Theory: Keynes's D-Z Model of Chapters 20 and 21

F. Modigliani presented a special case of Keynes’s General Theory result in 1944 in his “Liquidity Preference and the Theory of Interest and Money”. Modigliani sought to provide the IS-LM model of Hicks’s 1937 Econometrica interpretation of Keynes’s chapter 15 IS-LM model with microeconomic foundations in the theory of the firm that included a production function and labor market. Modigliani overlooked the fact that Keynes had already done exactly that in his chapters 20 and 21 of the General Theory. Section 4 of Keynes’s chapter 15 was the bridge connecting chapter 15 to chapters 20 and 21. Modigliani erred, however, in four ways. First, he used the theory of perfect competition, with its assumptions of perfect information and perfect prediction, and not Keynes’s theory of pure competition. Second, Keynes defined p to be an expected price in the General Theory, whereas Modigliani defined his capital P to be an actual price. This led to his third mistake, which was to define the necessary and sufficient first and second order conditions for optimality, leading to a profit maximum, in the labor market, given decreasing returns, as being where the ACTUAL real wage of labor equaled the marginal productivity of labor. Keynes’ condition is that it is the EXPECTED real wage of labor that equals the marginal productivity of labor. This leads directly to Keynes’s Aggregate Supply Curve of multiple equilibria, which is a locus of the entire set of all possible D-Z intersections, which will lead to one Y value, whereas Modigliani is stuck with only one equilibrium. Modigliani thus has the equivalent of Keynes’s Y-multiplier income expenditure model result from chapter 10 of the General Theory, but no D-Z model of expected prices and expected profits from chapters 20 and 21 of the General Theory. Modigliani’s fourth mistake was that he replaced Keynes’s uncertainty, a function of the weight of the evidence, with risk. This follows from Modigliani’s acceptance of the de Finetti subjective theory of probability, where there is only risk and no uncertainty because all probabilities must be additive, precise probabilities, whereas for Keynes most probabilities must be non-additive, imprecise or indeterminate interval valued probabilities. Modigliani’s paper thus becomes a special case of Keynes’s General Theory analysis in chapters 20 and 21.

Real-Time Optimization via Modifier Adaptation using Partial Plant Models

Modifier adaptation is a real-time optimization method that has the ability to reach the plant optimum upon convergence despite the presence of uncertainty in the form of plant-model mismatch and disturbances. The approach is based on modifying the cost and constraint functions predicted by the model by means of appropriate first-order correction terms. The main difficulty lies in the fact that these correction terms require the plant cost and constraint gradients to be estimated from experimental data at each iteration. Although the model used can support a significant level of approximation, it must satisfy the following two requirements: (i) a model adequacy condition related to the second-order optimality conditions must be valid at the plant optimum, and (ii) the model must have the same input variables as the plant. In this paper, we consider the case where (ii) is not verified because only a partial or incomplete model is available. We propose to approximate the unmodeled part of the system by a linear model that is identified using the same excitation that is used in modifier adaptation for gradient estimation. The approach is illustrated through the simulated example of a reaction-separation system with recycle.

Learning optimal spatially-dependent regularization parameters in total variation image denoising

We consider a bilevel optimization approach in function space for the choice of spatially dependent regularization parameters in TV image denoising models. First- and second-order optimality conditions for the bilevel problem are studied when the spatially-dependent parameter belongs to the Sobolev space . A combined Schwarz domain decomposition-semismooth Newton method is proposed for the solution of the full optimality system and local superlinear convergence of the semismooth Newton method is verified. Exhaustive numerical computations are finally carried out to show the suitability of the approach.

An Extension of Yuan’s Lemma and Its Applications in Optimization

We prove an extension of Yuan's Lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of [A. Baccari and A. Trad. On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Opt., 15(2):394--408, 2005], where the classical necessary second-order optimality condition is proved under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the hessian of the Lagrangian evaluated at the vertices of the Lagrange multiplier set is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the jacobian matrix.

New Second-Order Karush–Kuhn–Tucker Optimality Conditions for Vector Optimization

In the present paper, we focus on the vector optimization problems with constraints, where objective functions and constrained functions are Frechet differentiable, and whose gradient mapping is locally Lipschitz. By using the second-order symmetric subdifferential and the second-order tangent set, we introduce some new types of second-order regularity conditions in the sense of Abadie. Then we establish some second-order necessary optimality conditions Karush–Kuhn–Tucker-type for local efficient (weak efficient, Geoffrion properly efficient) solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution to such problem.

Second-order sequence-based necessary optimality conditions in constrained nonsmooth vector optimization and applications

Several notions of sequential directional derivatives and sequential local approximations are introduced. Under (first-order) Hadamard differentiability assumptions of the data at the point of study, these concepts are utilized to analyze second-order necessary optimality conditions, which rely on given sequences, for local weak solutions in nonsmooth vector optimization problems with constraints. Some applications to minimax programming problems are also derived.

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

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

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.

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(⋅).

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.

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 ...

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

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.

Generalized parameter-free duality models in discrete minmax fractional programming based on second-order optimality conditions

In this paper, we construct six generalized second-order parameter-free duality models, and prove several weak, strong, and strict converse duality theorems for a discrete minmax fractional programming problem using two partitioning schemes and various types of generalized second-order $$(\mathcal {F},\beta ,\phi ,\rho ,\theta ,m)$$ -univexity (more compactly, ’second-order univexity’ is referred to as ’sounivexity’) assumptions. The obtained results are new and generalize most of results on discrete minmax fractional programming involving the second-order invexity as well as on second-order univexity in the literature.

On Necessary Optimality Conditions for Nonsmooth Vector Optimization Problems with Mixed Constraints in Infinite Dimensions

In this note, we develop first- and second-order necessary optimality conditions for local weak solutions in nonsmooth vector optimization problems subject to mixed constraints in infinite-dimensional settings. To this aim, we use some set-valued directional derivatives of the Hadamard type and tangent sets, and impose (first-order) Hadamard differentiability assumptions of the data at the point of consideration.