Existence and continuity of an implicit function in the neighborhood of an abnormal point

Solutions to the equation F(x, ??) = 0 with unknown x and the parameter ?? in the neighborhood of the solution (x _*, ??_*) under the additional constraint x ?? U, where U is a closed convex set, are studied. The sufficient conditions for existence of an implicit function without prior assumption of the normalcy of point x _* are given. The obtained result is used to investigate the local solvability of controlled systems with mixed constraints.     

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

Coincidence points of mappings in vector metric spaces with applications to differential equations and control systems

We prove a theorem on the coincidence points of two mappings acting on spaces equipped with a vector metric. By way of application, we obtain sufficient conditions for the existence of a solution of an ordinary differential equation unsolved for the derivative of the unknown function and local solvability conditions for a control system with mixed constraints.     

A note on Jensen’s inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.     

The p-Factor-Lagrange Methods for Degenerate Nonlinear Programming

The paper presents a new approach to solving nonlinear programming (NLP) problems for which the strict complementarity condition (SCC), a constraint qualification (CQ), and a second-order sufficient condition (SOSC) for optimality are not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equalities. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient condition (SOSC) for optimality. At the same time, we give examples of the problems where the SOSC does not hold, but our assumption is satisfied. Moreover, no estimation of the set of active constraints is required. The proposed approach can be applied to a variety of problems.     

Isolated calmness of solution mappings in convex semi-infinite optimization

This paper is concerned with isolated calmness of the solution mapping of a parameterized convex semi-infinite optimization problem subject to canonical perturbations. We provide a sufficient condition for isolated calmness of this mapping. This sufficient condition characterizes the strong uniqueness of minimizers, under the Slater constraint qualification. Moreover, on the assumption that the objective function and the constraints are linear, we show that this condition is also necessary for isolated calmness.     

On Jensen’s inequality for g-expectation and for nonlinear expectation

In this paper, we give a necessary and sufficient condition for g under which Jensen’s inequality holds for g-expectation. In particular, we show that if Jensen’s inequality holds for g-expectation, then g is independent of y and g is superhomogeneous. We also establish a necessary and sufficient condition under which Jensen’s inequality holds for a general filtration-consistent nonlinear expectation. Received: 18 January 2005     

Sensitivity analysis and the adjoint update strategy for an optimal control problem with mixed control-state constraints

In this article, an optimal control problem subject to a semilinear elliptic equation and mixed control-state constraints is investigated. The problem data depends on certain parameters. Under an assumption of separation of the active sets and a second-order sufficient optimality condition, Bouligand-differentiability (B-differentiability) of the solutions with respect to the parameter is established. Furthermore, an adjoint update strategy is proposed which yields a better approximation of the optimal controls and multipliers than the classical Taylor expansion, with remainder terms vanishing in L ^∞.     

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive necessary second-order optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients withrespect to the control of active mixed constraints are linearly independent, the necessary conditions follows from a Pontryagin minimum in the problem. Together with sufficient second-order conditions [70], the necessary conditions of the present paper constitute a pair of no-gap conditions.     

Stabilized SQP revisited

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47–73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, for the problem with equality constraints only, we prove superlinear convergence of sSQP under the assumption that the dual starting point is close to a noncritical multiplier. Since noncritical multipliers include all those satisfying SOSC but are not limited to them, we believe this gives the first superlinear convergence result for any Newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than SOSC. In the general case when inequality constraints are present, we show that such a relaxation of assumptions is not possible. We also consider applying sSQP to the problem where inequality constraints are reformulated into equalities using slack variables, and discuss the assumptions needed for convergence in this approach. We conclude with consequences for local regularization methods proposed in (Izmailov and Solodov SIAM J Optim 16:210–228, 2004; Wright SIAM J. Optim. 15:673–676, 2005). In particular, we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally.     

A generalized mixed vector variational-like inequality problem

In this paper, we introduce relaxed η-α-P-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η-complete semicontinuity and η-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

Second-Order Sufficient Conditions for State-Constrained Optimal Control Problems

Second-order sufficient optimality conditions (SSC) are derived for an optimal control problem subject to mixed control-state and pure state constraints of order one. The proof is based on a Hamilton-Jacobi inequality and it exploits the regularity of the control function as well as the associated Lagrange multipliers. The obtained SSC involve the strict Legendre-Clebsch condition and the solvability of an auxiliary Riccati equation. They are weakened by taking into account the strongly active constraints.     

Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation, namely, a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second-order optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second-order sufficient condition holds, we give a formula for the second-order expansion of the value of the problem as well as the directional derivative of the optimal control, when the cost function is perturbed. Then we extend the theory of second-order optimality conditions to the case of vector-valued controls when the feasible set is defined by local and smooth convex constraints. When the space dimension n is greater than 3, the results are based on a two norms approach, involving spaces L ² and L ^s , with s>n/2 . Accepted 27 January 1997     

Mathematical Programs with Semidefinite Cone Complementarity Constraints: Constraint Qualifications and Optimality Conditions

The tangent cone of gph $N_{S^n_+}$ plays an important role in developing necessary conditions for mathematical programs with semidefinite cone complementarity constraints. We demonstrate an elegant formula for the tangent cone of gph $N_{S^n_+}$ , based on which the Bouligand stationary point is characterized explicitly. The relationships among different stationary points under certain constraint qualifications are discussed. Then we propose a second order sufficient condition which can be weakened under the strict complementarity condition. Importantly, for the sake of algorithm design, under the assumption of strict complementarity condition, we give a nonsmooth equation reformulation of the stationary point, whose smoothing system is verified to be nonsingular at the stationary point under the proposed second order sufficient condition.     

A primal mixed formulation for exterior transmission problems in ${\bf R}^2$

Summary.   We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints. We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h). Received August 25, 1998 / Revised version received March 8, 2000 / Published online October 16, 2000     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

On the optimal control problems with Gâteaux differentiable operator constraints

In this paper the problem of optimal control with mixed equality and inequality operator constraints is considered under the assumption of Gâteaux differentiability. The extremum principle for this kind of problem is obtained by using some specification of the Dubovitskii-Milyutin method for the case of n equality constraints given by Gâteaux differentiable operators.     

Best mean-quasiconformal mappings

We look for best mean-quasiconformal mappings as extremals of the functional equal to the integral of the square of the functional of the conformality distortion multiplied by a special weight. The mapping inverse to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the Petrovskii ellipticity of the system of Euler equations for an extremal. We prove the local unique solvability of boundary values problems for this system in the 2-dimensional case. In the general case we prove the unique solvability of boundary value problems for the system linearized at the identity mapping.     

On the Unique Solvability of an Inverse Problem for Parabolic Equations under a Final Overdetermination Condition

We consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition. We obtain two types of conditions that are sufficient for the local solvability of the inverse problem and also prove the so-called Fredholm solvability of the inverse problem under study.