This paper deals with the frame topology optimization under the frequency constraint and proposes an algorithm that solves a sequence of relaxation problems to obtain a local optimal solution with high quality. It is ...
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This paper deals with the frame topology optimization under the frequency constraint and proposes an algorithm that solves a sequence of relaxation problems to obtain a local optimal solution with high quality. It is known that an optimal solution of this problem often has multiple eigenvalues and the feasible set is disconnected. Due to these two difficulties, conventional nonlinearprogramming approaches often converge to a local optimal solution that is unacceptable from a practical point of view. In this paper, we formulate the frequency constraint as a positive semidefinite constraint of a certain symmetric matrix, and then relax this constraint to make the feasible set connected. The proposed algorithm solves a sequence of the relaxation problems with gradually decreasing the relaxation parameter. The positive semidefinite constraint is treated with the logarithmic barrier function and, hence, the algorithm finds no difficulty in multiple eigenvalues of a solution. Numerical experiments show that global optimal solutions, or at least local optimal solutions with high qualities, can be obtained with the proposed algorithm.
This paper proposes a filter method for solving nonlinear semidefinite programming problems. Our method extends to this setting the filter SQP (sequential quadratic programming) algorithm, recently introduced for solv...
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This paper proposes a filter method for solving nonlinear semidefinite programming problems. Our method extends to this setting the filter SQP (sequential quadratic programming) algorithm, recently introduced for solving nonlinearprogramming problems, obtaining the respective global convergence results.
Recently, Chan and Sun [Chan, Z. X., D. Sun. Constraint nondegeneracy, strong regularity and nonsingularity in semidefiniteprogramming. SIAM J. Optim. 19 370-376.] reported for semidefiniteprogramming (SDP) that the...
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Recently, Chan and Sun [Chan, Z. X., D. Sun. Constraint nondegeneracy, strong regularity and nonsingularity in semidefiniteprogramming. SIAM J. Optim. 19 370-376.] reported for semidefiniteprogramming (SDP) that the primal/dual constraint nondegeneracy is equivalent to the dual/primal strong second order sufficient condition (SSOSC). This result is responsible for a number of important results in stability analysis of SDP. In this paper, we study duality of this type in nonlinear semidefinite programming (NSDP). We introduce the dual SSOSC at a Karush-Kuhn-Tucker (KKT) triple of NSDP and study its various characterizations and relationships to the primal nondegeneracy. Although the dual SSOSC is nothing but the SSOSC for the Wolfe dual of the NSDP, it suggests new information for the primal NSDP. For example, it ensures that the inverse of the Hessian of the Lagrangian function exists at the KKT triple and the inverse is positive definite on some normal space. It also ensures the primal nondegeneracy. Some of our results generalize the corresponding classical duality results in nonlinearprogramming studied by Fujiwara et al. [Fujiwara, O., S.-P. Han, O. L. Mangasarian. 1984. Local duality of nonlinear programs. SIAM J. Control Optim. 22 162-169]. For the convex quadratic SDP (QSDP), we have complete characterizations for the primal and dual SSOSC. Our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual SSOSC but also the primal and dual nondegeneracy at its solution, suggesting that it is a well-conditioned QSDP.
We analyze the rate of local convergence of the augmented Lagrangian method in nonlinearsemidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singul...
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We analyze the rate of local convergence of the augmented Lagrangian method in nonlinearsemidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and variational analysis on the projection operator in the symmetric matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold (c) over bar > 0.
This paper proposes a primal-dual interior-point filter method for nonlinear semidefinite programming, which is an extension of the work of Ulbrich et al. (Math. Prog., 100(2):379-410, 2004). We use a mixed norm to ta...
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This paper proposes a primal-dual interior-point filter method for nonlinear semidefinite programming, which is an extension of the work of Ulbrich et al. (Math. Prog., 100(2):379-410, 2004). We use a mixed norm to tackle with trust region constraints and global convergence to first-order critical points can be proved.
We consider the solution of nonlinear programs with nonlinearsemidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear sem...
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We consider the solution of nonlinear programs with nonlinearsemidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinearsemidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefiniteprogramming (SSP) method, which is a generalization of the well-known sequential quadratic programming method for standard nonlinear programs. We present a sensitivity result for nonlinearsemidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinearsemidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.
This article is a continuation of the paper Kovara and Stingl (Struct Multidisc Optim 33(4-5):323-335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems...
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This article is a continuation of the paper Kovara and Stingl (Struct Multidisc Optim 33(4-5):323-335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or "free sizing") and the free material optimization problems. We will present an efficient algorithm for solving large scale instances of these problems. Examples will demonstrate the efficiency of the algorithm and the importance of the local stress constraints. In particular, we will argue that in certain topology optimization problems, the addition of stress constraints must necessarily lead not only to the change of optimal topology but also optimal geometry. Contrary to that, in material optimization problems the stress singularity is treated by the change in the optimal material properties.
Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefin...
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Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty and augmented Lagrangian method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bezoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behaviour of PENNON.
We consider the duality theories in nonlinear semidefinite programming. Some duality theorems are established to show the important relations among the optimal solutions and optimal values of the primal, the dual and ...
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We consider the duality theories in nonlinear semidefinite programming. Some duality theorems are established to show the important relations among the optimal solutions and optimal values of the primal, the dual and the saddle point problems of nonlinear semidefinite programming. (c) 2005 Elsevier Ltd. All rights reserved.
The robustification of trading strategies is of particular interest in financial market applications. In this paper we robustify a portfolio strategy recently introduced in the literature against model errors in the s...
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The robustification of trading strategies is of particular interest in financial market applications. In this paper we robustify a portfolio strategy recently introduced in the literature against model errors in the sense of a worst case design. As it turns out, the resulting optimization problem can be solved by a sequence of linear and nonlinearsemidefinite programs (SDP/NSDP), where the nonlinearity is introduced by the parameters of a parabolic differential equation. The nonlinearsemidefinite program naturally arises in the computation of the worst case constraint violation which is equivalent to an eigenvalue minimization problem. Further we prove convergence for the iterates generated by the sequential SDP-NSDP approach.
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