Sequential Optimality Conditions Research Articles

New Sequential Optimality Conditions for Mathematical Programs with Complementarity Constraints and Algorithmic Consequences

In recent years, the theoretical convergence of iterative methods for solving nonlinear constrained optimization problems has been addressed using sequential optimality conditions, which are satisf...

Sequential characterizations of robust optimal solutions in uncertain convex programs via perturbation approach

By using robust optimization approach, necessary and sufficient sequential optimality conditions without any constraint qualifications for the general convex optimization problem in the face of data uncertainty are given in terms of the ε-subdifferential. A sequential condition involving only the subdifferentials is also derived using a version of the Brøndsted-Rockafellar theorem. Consequently, sequential Lagrange multiplier condition for robust optimal solution of convex optimization problem with cone constraints in the face of data uncertainty is given. It is worth pointing out that there is no compactness assumption of uncertainty set and upper semicontinuity of functions involved in our results.

A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasi-normality constraint qualification, which is associated with the external penalty theory....

Optimality conditions and global convergence for nonlinear semidefinite programming

Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush–Kuhn–Tucker (AKKT), well known in nonlinear optimization. The second one, called Trace-AKKT, is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided. We propose an augmented Lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed, weaker than previously considered ones, which are sufficient for the global convergence of the algorithm to a stationary point.

Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate property on the constraints holds at a point that satisfies a sequential optimality condition, such a point also satisfies the Karush-Kuhn-Tucker conditions. Those properties will be called strict constraint qualifications in this paper. As a consequence, for each sequential optimality condition, it is natural to ask for its weakest strict associated constraint qualification. This problem has been solved in a recent paper for the Approximate Karush-Kuhn-Tucker sequential optimality condition. In the present paper, we characterize the weakest strict constraint qualifications associated with other sequential optimality conditions that are useful for defining stopping criteria of algorithms. In addition, we prove all the implications between the new strict constraint qualifications and other (classical or strict) constraint qualifications.

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”

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.

Sequential optimality conditions for multiobjective fractional programming problems

In this paper, our aim is to develop necessary and sufficient optimality conditions in the absence of any constraint qualification for multiobjective fractional programming problem using the powerful combination of conjugate analysis and e-subdifferential calculus. Furthermore, as an application of these conditions we derive sequential duality results for this class of problems.

Sequential Optimality Conditions for Fractional Optimization with Applications to Vector Optimization

In this paper, in the absence of any constraint qualifications, a sequential Lagrange multiplier rule condition characterizing optimality for a fractional optimization problem is obtained in terms of the $$\varepsilon $$ ? -subdifferentials of the functions involved at the minimizer. The significance of this result is that it yields the standard Lagrange multiplier rule condition for the fractional optimization problem under a simple closedness condition that is much weaker than the well-known constraint qualifications. A sequential condition characterizing optimality involving only subdifferentials at nearby points to the minimizer is also investigated. As applications, the proposed approach is applied to investigate sequential optimality conditions for vector fractional optimization problems.

On the behaviour of constrained optimization methods when Lagrange multipliers do not exist

Sequential optimality conditions are related to stopping criteria for nonlinear programming algorithms. Local minimizers of continuous optimization problems satisfy these conditions without constraint qualifications. It is interesting to discover whether well-known optimization algorithms generate primal–dual sequences that allow one to detect that a sequential optimality condition holds. When this is the case, the algorithm stops with a ‘correct’ diagnostic of success (‘convergence’). Otherwise, closeness to a minimizer is not detected and the algorithm ignores that a satisfactory solution has been found. In this paper it will be shown that a straightforward version of the Newton–Lagrange (sequential quadratic programming) method fails to generate iterates for which a sequential optimality condition is satisfied. On the other hand, a Newtonian penalty–barrier Lagrangian method guarantees that the appropriate stopping criterion eventually holds.

On sequential optimality conditions for smooth constrained optimization

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate Karush–Kuhn–Tucker and approximate gradient projection conditions are analysed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed.

On sequential optimality conditions for smooth constrained optimization

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate Karush–Kuhn–Tucker and approximate gradient projection conditions are analysed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed.

A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences

Necessary first-order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications. A new condition of this type is introduced in the present paper. It is proved that a well-established augmented Lagrangian algorithm produces sequences whose limits satisfy the new condition. Practical consequences are discussed.

Sequential optimality conditions for composed convex optimization problems

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.

ON OPTIMALITY CONDITIONS FOR ABSTRACT CONVEX VECTOR OPTIMIZATION PROBLEMS

A sequential optimality condition characterizing the efficient solution without any constraint qualification for an abstract convex vector optimization problem is given in sequential forms using subdifferentials and <TEX>${\epsilon}$</TEX>-subdifferentials. Another sequential condition involving only the subdifferentials, but at nearby points to the efficient solution for constraints, is also derived. Moreover, we present a proposition with a sufficient condition for an efficient solution to be properly efficient, which are a generalization of the well-known Isermann result for a linear vector optimization problem. An example is given to illustrate the significance of our main results. Also, we give an example showing that the proper efficiency may not imply certain closeness assumption.

Sequential Lagrangian Conditions for Convex Programs with Applications to Semidefinite Programming

In this paper it is shown that, in the absence of any regularity condition, sequential Lagrangian optimality conditions as well as a sequential duality results hold for abstract convex programs. The significance of the results is that they yield the standard optimality and duality results for convex programs under a simple closed-cone condition that is much weaker than the well-known constraint qualifications. As an application, a sequential Lagrangian duality, saddle-point conditions, and stability results are derived for convex semidefinite programs.