The stabilized sequential quadratic programming (SQP) method can effectively deal with degenerate nonlinear optimization problems. In the case of nonunique Lagrange multipliers associated with a stationary point of an...
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The stabilized sequential quadratic programming (SQP) method can effectively deal with degenerate nonlinear optimization problems. In the case of nonunique Lagrange multipliers associated with a stationary point of an optimization problem, the stabilized SQP method still obtains superlinear and/or quadratic convergence to a primal-dual solution. In this paper, we propose a stabilized sequential quadratic semidefiniteprogramming method for degenerate nonlinear semidefinite programming problems. Under the local error bound condition, the strict complementarity condition, and the second-order sufficient condition, we establish superlinear and/or quadratic convergence of the proposed method.
We introduce a quadratically convergent semismooth Newton method for nonlinear semidefinite programming that eliminates the need for the generalized Jacobian regularity, a common yet stringent requirement in existing ...
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We introduce a quadratically convergent semismooth Newton method for nonlinear semidefinite programming that eliminates the need for the generalized Jacobian regularity, a common yet stringent requirement in existing approaches. Our strategy involves identifying a single nonsingular element within the Bouligand generalized Jacobian, thus avoiding the standard requirement for nonsingularity across the entire generalized Jacobian set, which is often too restrictive for practical applications. The theoretical framework is supported by introducing the weak second order condition (W-SOC) and the weak strict Robinson constraint qualification (W-SRCQ). These conditions not only guarantee the existence of a nonsingular element in the generalized Jacobian but also forge a primal-dual connection in linearly constrained convex quadratic programming. The theoretical advancements further lay the foundation for the algorithmic design of a novel semismooth Newton method, which integrates a correction step to address degenerate issues. Particularly, this correction step ensures the local convergence as well as a superlinear/quadratic convergence rate of the proposed method. Preliminary numerical experiments corroborate our theoretical findings and underscore the practical effectiveness of our method.
The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numer...
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The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numerous efficient solvers to improve numerical performance to solve many problems. In this paper, without requiring the uniqueness of multipliers, the local (asymptotic Q-superlinear) Q-linear convergence rate of the primal-dual sequences generated by ALM for the nonlinear semidefinite programming is established by assuming the second-order sufficient condition and the semi-isolated calmness of the Karush-Kuhn-Tucker solution under some mild conditions.
Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for nonpolyhedral problems remains...
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Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for nonpolyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second-order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP) without requiring the uniqueness of the multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under the strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semismooth Newton method to solve the ALM subproblems of NLSDP because the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.
We propose a line search exact penalty method with bi-object strategy for nonlinearsemidefinite *** each iteration,we solve a linear semidefiniteprogramming to test whether the linearized constraints are consistent ...
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We propose a line search exact penalty method with bi-object strategy for nonlinearsemidefinite *** each iteration,we solve a linear semidefiniteprogramming to test whether the linearized constraints are consistent or *** search direction is generated by a piecewise quadratic-linear model of the exact penalty *** penalty parameter is only related to the information of the current iterate *** line search strategy is a penalty-free *** and local convergence are analyzed under suitable *** finally report some numerical experiments to illustrate the behavior of the algorithm on various degeneracy situations.
In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems, in which the iterates converge to a point that satisfies the first-order and second-o...
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In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems, in which the iterates converge to a point that satisfies the first-order and second-order optimality conditions. The method consists of the outer iteration (SDPIP-revised) that finds a Karush-Kuhn-Tucker (KKT) point which satisfies the second-order optimality condition, and the inner iteration (SDPTR-revised) that calculates an approximate barrier KKT point. Algorithm SDPTR-revised uses a commutative class of Newton-like directions within the framework of the trust-region method in the primal-dual space. In addition, we also use a direction of negative curvature when it exists. The proposed algorithm employs a new method that generates negative-curvature directions in the existence of l(1)-type penalty term for equality constraints. It is proved that there exists a limit point of the generated sequence which satisfies the second-order optimality condition along with the barrier KKT conditions.
In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a Karush–Kuhn–Tucker ...
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In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a Karush–Kuhn–Tucker (KKT) point and the inner iteration (SDPTR) that calculates an approximate barrier KKT point. Algorithm SDPTR combines a commutative class of Newton-like directions with the steepest descent type direction within the framework of the trust-region strategy. We present a trust-region method in primal-dual space and prove the global convergence property of the proposed method. Some numerical experiments are given. In addition, we also present second-order approximations to the primal-dual merit function, and a trust-region method in primal space in Appendix.
Jordan Algebras are an important tool for dealing with semidefiniteprogramming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimali...
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Jordan Algebras are an important tool for dealing with semidefiniteprogramming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed;in particular, a global convergence result is available even when Lagrange multipliers are unbounded.
In this paper,we present a QP-free algorithm without a penalty function or a filter for nonlinearsemidefinite *** each iteration,two systems of linear equations with the same coefficient matrix are solved to determin...
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In this paper,we present a QP-free algorithm without a penalty function or a filter for nonlinearsemidefinite *** each iteration,two systems of linear equations with the same coefficient matrix are solved to determine search direction;the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently *** is no feasibility restoration phase in our algorithm,which is necessary for traditional filter *** proposed algorithm is globally convergent under some mild *** numerical results indicate that the proposed algorithm is comparable.
In this work, we present an Augmented Lagrangian algorithm for nonlinearsemidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinearprogramming. This method works with tw...
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In this work, we present an Augmented Lagrangian algorithm for nonlinearsemidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinearprogramming. This method works with two levels of constraints;one that is penalized and other that is kept within the subproblems. This is done to allow exploiting the subproblem structure while solving it. The global convergence theory is based on recent results regarding approximate Karush-Kuhn-Tucker optimality conditions for NLSDPs, which are stronger than the usually employed Fritz John optimality conditions. Additionally, we approach the problem of covering a given object with a fixed number of balls with a minimum radius, where we employ some convex algebraic geometry tools, such as Stengle's Positivstellensatz and its variations, which allows for a much more general model. Preliminary numerical experiments are presented.
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