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In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss–Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an inexact sGS method, instead of an unknown number of cycles, to solve each of the subproblems involved. With some simple and implementable error tolerance criteria, the cost for solving the subproblems can be greatly reduced, and many steps in the forward sweep of each sGS cycle can often be skipped, which further contributes to the efficiency of the proposed method. Global convergence as well as the iteration complexity in the non-ergodic sense is established. Preliminary numerical experiments on some high-dimensional linear and convex quadratic SDP problems with a large number of linear equality and inequality constraints are also provided. The results show that for the vast majority of the tested problems, the sGS-imsPADMM is 2–3 times faster than the directly extended multi-block ADMM with the aggressive step-length of 1.618, which is currently the benchmark among first-order methods for solving multi-block linear and quadratic SDP problems though its convergence is not guaranteed.

The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multi-variable real-valued function with the tensor T-product. Inspired by an equivalent characterization of a twice continuously T-differentiable multi-variable real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analogue to the widely used semidefinite programming (SDP for short), we introduce the semidefinite programming over the space of third-order symmetric tensors (T-semidefinite programming or TSDP for short), and provide a way to solve the TSDP problem by converting it into an SDP problem in the complex domain. Furthermore, we give several TSDP examples and especially some preliminary numerical results for two unconstrained polynomial optimization problems. Experiments show that finding the global minimums of polynomials via the TSDP relaxation outperforms the traditional SDP relaxation for the test examples.

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.

In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving degenerate nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Unlike the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems inexactly, and each QSDP is feasible. One more remarkable point of the proposed method is that constraint qualifications or boundedness of Lagrange multiplier sequences are not required in the global convergence analysis. Specifically, without assuming such conditions, we prove the global convergence to a point satisfying any of the following: the stationary conditions for the feasibility problem, the approximate-Karush–Kuhn–Tucker (AKKT) conditions, and the trace-AKKT conditions. Finally, we conduct some numerical experiments to examine the efficiency of the proposed method.

In this paper, we deal with exact semidefinite programming (SDP) reformulations for a class of adjustable robust quadratic optimization problems with affine decision rules. By virtue of a special semidefinite representation of the non-negativity of separable non-convex quadratic functions on box uncertain sets, we establish an exact SDP reformulation for this adjustable robust quadratic optimization problem on spectrahedral uncertain sets. Note that the spectrahedral uncertain set contains commonly used uncertain sets, such as ellipsoids, polytopes, and boxes. As special cases, we also establish exact SDP reformulations for this adjustable robust quadratic optimization problems when the uncertain sets are ellipsoids, polytopes, and boxes, respectively. As applications, we establish the corresponding results for fractionally adjustable robust quadratic optimization problems.

In this paper, we deal with a new class of SOS-convex (sum of squares convex) polynomial optimization problems with spectrahedral uncertainty data in both the objective and constraints. By using robust optimization and a weighted-sum scalarization methodology, we first present the relationship between robust solutions of this uncertain SOS-convex polynomial optimization problem and that of its corresponding scalar optimization problem. Then, by using a normal cone constraint qualification condition, we establish necessary and sufficient optimality conditions for robust weakly efficient solutions of this uncertain SOS-convex polynomial optimization problem based on scaled diagonally dominant sums of squares conditions and linear matrix inequalities. Moreover, we introduce a semidefinite programming relaxation problem of its weighted-sum scalar optimization problem, and show that robust weakly efficient solutions of the uncertain SOS-convex polynomial optimization problem can be found by solving the corresponding semidefinite programming relaxation problem.

The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem satisfies the strong duality relations and does not require any additional regularity assumptions such as constraint qualifications. The main difference with the previously obtained results consists in the fact that now the extended dual problem uses neither the immobile indices themselves nor the explicit information about the convex hull of these indices. The strong duality formulations presented in the paper for linear copositive problems have similar structure and properties as that proposed in the works by M. Ramana, L. Tuncel, and H. Wolkowicz, for semidefinite programming.

To address the problem of low indoor positioning accuracy in time-of-arrival systems in the non-line-of-sight (NLOS) environments, we proposed an optimized positioning algorithm based on semidefinite programming (SDP). This algorithm reduces the NLOS error through a novelty method. Compared with the original SDP algorithm, we optimized the algorithm’s objective function by avoiding its dependence on the prior information, thereby decreasing infeasibility problems. The experiment showed that the proposed algorithm’s accuracy is superior to that of the traditional SDP algorithm in the same indoor environment.

Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that only bounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.

Mehrotra and Özevin (SIAM J Optim 19:1846–1880, 2009) computationally found that a weighted barrier decomposition algorithm for two-stage stochastic conic programs achieves significantly superior performance when compared to standard barrier decomposition algorithms existing in the literature. Inspired by this motivation, Mehrotra and Özevin (SIAM J Optim 20:2474–2486, 2010) theoretically analyzed the iteration complexity for a decomposition algorithm based on the weighted logarithmic barrier function for two-stage stochastic linear optimization with discrete support. In this paper, we extend the aforementioned theoretical paper and its self-concordance analysis from the polyhedral case to the semidefinite case and analyze the iteration complexity for a weighted logarithmic barrier decomposition algorithm for two-stage stochastic convex quadratic SDP with discrete support.