Semidefinite Programming Problem Research Articles

Abstract This paper presents a robust framework for handling a conic multiobjective linear optimization problem, where the objective and constraint functions are involving affinely parameterized data uncertainties. More precisely, we examine optimality conditions and calculate efficient solutions of the conic robust multiobjective linear problem. We provide necessary and sufficient linear conic criteria for efficiency of the underlying conic robust multiobjective linear program. It is shown that such optimality conditions can be expressed in terms of linear matrix inequalities and second-order conic conditions for a multiobjective semidefinite program and a multiobjective second order conic program, respectively. We show how efficient solutions of the conic robust multiobjective linear problem can be found via its conic programming reformulation problems including semidefinite programming and second-order cone programming problems. Numerical examples are also provided to illustrate that the proposed conic programming reformulation schemes can be employed to find efficient solutions for concrete problems including those arisen from practical applications.

With access to new energy sources, the problem of reactive power optimization and dispatching has become increasingly important for research. However, the reactive power optimization problem is a mixed integer nonlinear optimization problem. In order to solve the integer variables and nonlinear conditions existing therein, a method for coordinated reactive power optimization and dispatching based on semidefinite programming is proposed. Firstly, a reactive power optimization model considering discrete variables and continuous variables is established with the minimization of total operating cost as the objective function; secondly, the discrete variables are transformed into equality constraints by quadratic equations, and then a solvable semi-definite programming problem is obtained; thirdly, the rank-one constraint is restored by the Iterative Optimization based Gaussian Randomization Method (IOGRM), and the optimal solution equivalent to the original problem is obtained. Finally, the correctness and effectiveness of the proposed model and solution method are verified by analyzing and comparing with the second-order cone programming (SOCP) through the modified IEEE standard example.

This paper focuses on two-dimensional (2-D) direction-of-arrival (DOA) estimation for an L-shaped array. While recent studies have explored sparse methods for this problem, most exploit only the cross-correlation matrix, neglecting self-correlation information and resulting accuracy degradation. We propose a multidimensional matrix completion method that employs joint sparsity and redundant correlation information embedded in the covariance matrix to reconstruct a structured matrix compactly coupling the two DOA parameters. A semidefinite program problem formulated via covariance fitting criteria is proved equivalent to the atomic norm minimization framework. The alternating direction method of multipliers is designed to reduce computational costs. Numerical results corroborate the analysis and demonstrate the superior estimation accuracy, identifiability, and resolution of the proposed method.

Lovász theta SDP problem can often be regarded as semidefinite programming (SDP) relaxation of combinatorial optimization problems in graph theory and appears in various fields. In this paper, we study a semismooth Newton based augmented Lagrangian (Ssnal) algorithm for solving Lovász theta SDP problem. There are three major ingredients in this paper. Firstly, we design efficient implementations of the Ssnal algorithm for solving dual problem of Lovász theta SDP problem. Secondly, the global convergence and local asymptotic superlinear convergence of the Ssnal algorithm are characterized under very mild conditions, in which a semismooth Newton (Ssn) method with superlinear or even quadratic convergence is applied to solve the subproblem. Finally, numerical experiments conducted on random and real data sets demonstrate that the Ssnal algorithm outperforms Sdpnal+ solver. In particular, the Ssnal algorithm can efficiently solve large-scale Lovász theta SDP problems with high accuracy, where the matrix dimension is up to 3000 and the number of equality constraints is up to 27,484.

Log-det semidefinite programming (SDP) problems are optimization problems that often arise from Gaussian graphical models. A log-det SDP problem with an ℓ 1 -norm term has been examined in many methods, and the dual spectral projected gradient (DSPG) method by Nakagaki et al. in 2020 is designed to efficiently solve the dual problem of the log-det SDP by combining a non-monotone line-search projected gradient method with the step adjustment for positive definiteness. In this paper, we extend the DSPG method for solving a generalized log-det SDP problem involving additional terms to cover more structures in Gaussian graphical models in a unified style. We establish the convergence of the proposed method to the optimal value. We conduct numerical experiments to illustrate the efficiency of the proposed method.

LoRADS is an enhanced first-order method solver for low rank Semi-definite programming problems (SDPs). LoRADS is written in ANSI C and is maintained by Cardinal Operations COPT development team. More features are still under active development.

Abstract We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art semidefinite programming solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.

To enhance the signature of an extended target in a SAR image, a robust waveform design method is presented for spectrally dense environments. First, the problem is formulated by maximizing the worst-case signal-to-clutter ratio (SCR) over the uncertainty set of statistics for both target and background scattering characteristics, subject to energy, similarity, and spectrum constraints. Second, the closed-form solutions for the uncertain statistics are derived. The problem of maximizing worst-case SCR is boiled down to a nonconvex fractional quadratically constrained quadratic problem (QCQP). Resorting to the Dinkelbach’s algorithm and Lagrange duality, the QCQP is split into a series of solvable semidefinite programming problems. A convergence analysis is conducted, where a sufficient condition for global convergence is derived. Finally, numerical examples are presented to demonstrate the performance of the proposed scheme.

Abstract In this paper, we study the Aubin property of the Karush-Kuhn-Tucker solution mapping for the nonlinear semidefinite programming (NLSDP) problem at a locally optimal solution. In the literature, it is known that the Aubin property implies the constraint nondegeneracy by Fusek (SIAM J. Optim. 23:1041–1061, 2013) and the second-order sufficient condition by Ding et al. (SIAM J. Optim. 27:67–90, 2017). Based on the Mordukhovich criterion, here we further prove that the strong second-order sufficient condition is also necessary for the Aubin property to hold. Consequently, several equivalent conditions including the strong regularity are established for NLSDP’s Aubin property. Together with the recent progress made by Chen et al. (SIAM J. Optim. 35:712–738, 2025) on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming, this paper constitutes a significant step forward in characterizing the Aubin property for general non-polyhedral $$C^2$$ C 2 -cone reducible constrained optimization problems.

The problem of semi-definite programming (SDP) extends linear programming (LP) to solve a broader range of optimization problems, with significant advancements in algorithmic methods, particularly interior point techniques. In this article, we a logarithmic penalty approach for resolving SDP problems, where the direction of descent is determined using Newton's method. Additionally, for the step length, we propose new, more efficient, and robust lower bound functions. These proposed functions improve the accuracy and efficiency of the solution process. The effectiveness of the method is demonstrated through extensive numerical simulations, which validate the claims made in this study. The results confirm the practical feasibility and performance of the approach in solving complex semi-definite programming problems.

This article addresses the optimal scheduling problem for linear deception attacks in multi-channel cyber–physical systems. The scenario where the attacker can only attack part of the channels due to energy constraints is considered. The effectiveness and stealthiness of attacks are quantified using state estimation error and Kullback–Leibler divergence, respectively. Unlike existing strategies relying on zero-mean Gaussian distributions, we propose a generalized attack model with Gaussian distributions characterized by time-varying means. Based on this model, an optimal stealthy attack strategy is designed to maximize remote estimation error while ensuring stealthiness. By analyzing correlations among variables in the objective function, the solution is decomposed into a semi-definite programming problem and a 0–1 programming problem. This approach yields the modified innovation and an attack scheduling matrix. Finally, numerical simulations validate the theoretical results.

HDSDP is a numerical software solving semidefinite programming problems. The main framework of HDSDP resembles the dual-scaling interior point solver DSDP [Benson and Ye, 2008] and several new features, including a dual method based on the simplified homogeneous self-dual embedding, have been implemented. The embedding technique enhances the stability of the dual method , and several new heuristics and computational techniques are designed to accelerate its convergence. HDSDP aims to show how the dual-scaling algorithm benefits from the self-dual embedding, and it is developed in parallel to DSDP5.8. Numerical experiments over several classical benchmark datasets exhibit its robustness and efficiency, particularly its advantages on SDP instances featuring low-rank structure and sparsity. HDSDP is open-sourced under an MIT license and available at https://github.com/Gwzwpxz/HDSDP .

In this paper, we analyze conic minimax convex polynomial optimization problems. Under a suitable regularity condition, an exact conic programming relaxation is established based on a positivity characterization of a max function over a conic convex system. Further, we consider a general conic minimax ρ-convex polynomial optimization problem, which is defined by appropriately extending the notion of conic convexity of a vector-valued mapping. For this problem, it is shown that a Karush-Kuhn-Tucker condition at a global minimizer is necessary and sufficient for ensuring an exact relaxation with attainment of the conic programming relaxation. The exact conic programming relaxations are applied to SOS-convex polynomial programs, where appropriate choices of the data allow the associated conic programming relaxation to be reformulated as a semidefinite programming problem. In this way, we can further elaborate the obtained results for other special settings including conic robust SOS-convex polynomial problems and difference of SOS-convex polynomial programs.

This work investigates the possibility to improve the computational efficiency of a set-based method for the trajectory planning of a car-like vehicle through artificial intelligence. Planning is performed on a graph that represents the operating scenario in which the vehicle moves, and the kinodynamic feasibility of the trajectories is guaranteed through a series of set-based arguments, which involve the solution of semi-definite programming problems. Navigation in the graph is performed through a hybrid A* algorithm whose performance metrics are improved through a properly trained classificator, which can forecast whether a candidate trajectory segment is feasible or not. The proposed solution is validated through numerical simulations, with a focus on the effects of different classificators features and by using two different kinds of artificial intelligence: a support vector machine (SVM) and a long-short term memory (LSTM). Results show up to a 28% reduction in computational effort and the importance of lowering the false negative rate in classification for achieving good planning performance outcomes.

ABSTRACTScheduling plans catering to dynamic and complex passenger demands have drawn recent attention. Given dynamic demand, there is an urgent need to explore methods for extracting valid data from vast amounts of information and achieving flexible, robust parametric control of scheduling to boost transportation resource utilization efficiency. This paper proposes a deep learning technique to construct uncertainty sets using first‐ and second‐order moment information of passenger demand. Based on previous research under deterministic demands, a distributional robust optimization model for integrated line plan and timetable scheduling is established. Unlike other robust optimization models, the distributional robust one can better utilize the information in uncertain data. To handle the ensuing mixed integer semidefinite programming problem, a generalized benders decomposition algorithm post‐linearization is presented, which decomposes the model for iterative solving. Notably, the proposed model attains an average demand satisfaction rate 10.19% higher than the deterministic demand model, reduces train usage by 9, and lifts the average full load rate by 19.05% compared to the strongly robust model. It can flexibly select parameters for diverse demand scenarios and decision‐making objectives, offering theoretical support for planning under uncertain passenger demands.

We consider the qualitative and quantitative stability of parametric semidefinite programming problems which are more general than C 2 -smooth parameterization. We do not require the differentiability of problem functions. We show that under metric regularity and Lipschitz-like property of problem functions, the optimal solution set and optimal value function of a restricted perturbed problem are upper semicontinuous, and the modulus of upper semicontinuity for the solution set can be described by a growth function. In addition, we prove that the 1/ k -order Hölder stability of the restricted optimal solution set can be obtained by adding the k -order growth condition of the unperturbed problem and illustrating its applications with examples. Especially, when k = 1, the Hölder stability is just the calmness.

Geometry optimization was recently introduced to existing truss topology optimization with global stability constraints. The resulting problems are formulated as highly nonlinear semidefinite programming problems that demand extensive computational effort to solve and have been solved only for small problem instances. The main challenge for effective computation is the positive semidefinite constraints which involve large sparse matrices. In this paper, we apply several techniques to tackle the challenge. First, we use the well-known chordal decomposition approach to replace each positive semidefinite constraint on a large sparse matrix by several positive semidefinite constraints on smaller submatrices together with suitable linking constraints. Moreover, we further improve the efficiency of the decomposition by applying a graph-based clique merging strategy to combine submatrices with significant overlap. Next, we couple these techniques with an optimization algorithm that fully exploits the structure of the smaller submatrices. As a result, we can solve much larger problems, which allows us to extend the existing single-load case to the multiple-load case, and to provide a computationally tractable approach for the latter case. Finally, we employ adaptive strategies from previous studies to solve successive problem instances, enabling the joints to navigate larger regions, and ultimately obtain further improved designs. The efficiency of the overall approach is demonstrated via computational experiments on large problem instances.

In this paper, we consider the nonconvex extended trust-region subproblem with two intersecting linear inequality constraints, (ETR 2 ), and use a sequence of semi-definite programming (SDP) problems with second-order-cone(SOC) constraints to eliminate the duality gap of the SOC reformulation for (ETR 2 ). We first narrow the duality gap of the SOC reformulation by adding a new appropriate SOC constraint, and a sufficient condition is presented to characterize when the new SOC constraint is valid. Then we establish an iterative algorithm and the results of numerical experiments show that the iterative algorithm works efficiently in eliminating the SDPR-SOCR gap of (ETR 2 ).

This paper proposes a squared smoothing Newton method via the Huber smoothing function for solving semidefinite programming problems (SDPs). We first study the fundamental properties of the matrix-valued mapping defined upon the Huber function. Using these results and existing ones in the literature, we then conduct rigorous convergence analysis and establish convergence properties for the proposed algorithm. In particular, we show that the proposed method is well-defined and admits global convergence. Moreover, under suitable regularity conditions, that is, the primal and dual constraint nondegenerate conditions, the proposed method is shown to have a super-linear convergence rate. To evaluate the practical performance of the algorithm, we conduct extensive numerical experiments for solving various classes of SDPs. Comparison with the state-of-the-art SDP solvers demonstrates that our method is also efficient for computing accurate solutions of SDPs. Funding: The research of D. Sun was supported in part by the Hong Kong Research Grants Council under Grant 15307523, and the research of K.-C. Toh was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [MOE-2019-T3-1-010].

A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals