Semidefinite Optimization Problems Research Articles

Computing the recession cone of a convex upper image via convex projection

AbstractIt is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.

Quantum Interior Point Methods for Semidefinite Optimization

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem n, but has worse dependence on other numerical parameters.

Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function that covers most of the general classes of probabilistic constraints: [Formula: see text]where [Formula: see text] is a convex cone of a Banach space. The conic inclusion [Formula: see text] represents an abstract system of inequalities, and ξ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function [Formula: see text]. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can, thus, entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite, and probust chance-constrained optimization problems. Funding: P. Pérez-Aros was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular [Grants 1200283 and 1190110] and Fondecyt Exploración [Grant 13220097]. C. Soto was supported by the National Agency for Research and Development (ANID)/Scholarship Program/Doctorado Nacional Chile [Grant 2017-21170428]. E. Vilches was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and Fondecyt Regular [Grant 1200283] and Fondecyt Exploración [Grant 13220097] from ANID-Chile.

Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

This paper introduces a 3D equilibrium-based finite element formulation for reinforced concrete. The stress unknowns in the proposed formulation strongly satisfy the equilibrium equations throughout the entire volume, with the exception of the mesh interfaces where the rebars intersect. The concrete stress tensor is interpolated using tetrahedral piece-wise linear elements with statically admissible discontinuities, while the rebars are considered as 1D elements embedded into the concrete, intersecting the triangular mesh. Thus, the global equilibrium is ensured by writing for each face of the mesh the traction continuity equation, while including the rebar stress contribution for the intersected triangular faces. The elastic equilibrium formulation is then written into an equivalent optimization problem, extended to the elastoplastic case by simply adding semi-definite matrix constraints on the concrete stress tensor, corresponding to a Rankine or a truncated Mohr–Coulomb criterion. As for the rebars, they are considered to obey a 1D perfectly elastoplastic behavior. The present formulation is also developed for limit analysis, to directly obtain an estimate of the lower bound of the collapse load. The resulting semi-definite programming optimization problems are solved using an in–house interior point algorithm.

Block Factor-Width-Two Matrices and Their Applications to Semidefinite and Sum-of-Squares Optimization

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a trade-off between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings.

Cardinality-constrained distributionally robust portfolio optimization

This paper studies a distributionally robust portfolio optimization model with a cardinality constraint for limiting the number of invested assets. We formulate this model as a mixed-integer semidefinite optimization (MISDO) problem by means of the moment-based ambiguity set of probability distributions of asset returns. To exactly solve large-scale problems, we propose a specialized cutting-plane algorithm that is based on bilevel optimization reformulation. We prove the finite convergence of the algorithm. We also apply a matrix completion technique to lower-level SDO problems to make their problem sizes much smaller. Numerical experiments demonstrate that our cutting-plane algorithm is significantly faster than the state-of-the-art MISDO solver SCIP-SDP. We also show that our portfolio optimization model can achieve good investment performance compared with the conventional robust optimization model based on the ellipsoidal uncertainty set.

LMI-based Data-Driven Robust Model Predictive Control

Predictive control, which is widely used currently, bases on a model of the system to compute the applied input optimizing the future system behavior. If the nominal models are not given or are uncertain, data-driven model predictive control approaches can be employed, where control input is directly obtained from past measured trajectories. Using a data informativity framework and Finsler's lemma, we propose a data-driven robust linear matrix inequality-based model predictive control scheme that considers input and state constraints for linear parameter-varying systems and Lur'e-type nonlinear systems. Using these data, we formulate the problem as a semi-definite optimization problem, whose solution provides the matrix gain for the linear feedback, while the decisive variables are independent of the length of the measurement data. The designed controller stabilizes the closed-loop system asymptotically and guarantees constraint satisfaction. Numerical examples are conducted to illustrate the method.

Measurement-Based Control for Minimizing Energy Functions in Quantum Systems

In variational quantum algorithms (VQAs), the most common objective is to find the minimum energy eigenstate of a given energy Hamiltonian. In this paper, we consider the general problem of finding a sufficient control Hamiltonian structure that, under a given feedback control law, ensures convergence to the minimum energy eigenstate of a given energy function. By including quantum non-demolition (QND) measurements in the loop, convergence to a pure state can be ensured from an arbitrary mixed initial state. Based on existing results on strict control Lyapunov functions, we formulate a semidefinite optimization problem, whose solution defines a non-unique control Hamiltonian, which is sufficient to ensure almost sure convergence to the minimum energy eigenstate under the given feedback law and the action of QND measurements. A numerical example is provided to showcase the proposed methodology.

A revised sequential quadratic semidefinite programming method for nonlinear semidefinite optimization

In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic semidefinite programming (SQSDP) method for solving, in particular, degenerate nonlinear semidefinite optimization problems. The algorithm is shown to converge globally without a constraint qualification, and it has some nice properties, including the feasible subproblems, and their possible inexact computations. In particular, the convergence was established for approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two sequential optimality conditions for the nonlinear conic contexts. However, recently, complementarity-AKKT (CAKKT) conditions were also considered, as an alternative to the previous mentioned ones, that is more practical. Since few methods are shown to converge to CAKKT points, at least in conic optimization, and to complete the study associated to the SQSDP, here we propose a revised version of the method, maintaining the good properties. We modify the previous algorithm, prove the global convergence in the sense of CAKKT, and show some preliminary numerical experiments.

Strong duality in parametric robust semi-definite linear programming and exact relaxations

This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.

A class of new search directions for full-NT step feasible interior point method in semidefinite optimization

In this paper, based on Darvay et al.’s strategy for linear optimization (LO) (Z. Darvay and P.R. Takács, Optim. Lett. 12 (2018) 1099–1116.), we extend Kheirfam et al.’s feasible primal-dual path-following interior point algorithm for LO (B. Kheirfam and A. Nasrollahi, Asian-Eur. J. Math. 1 (2020) 2050014.) to semidefinite optimization (SDO) problems in order to define a class of new search directions. The algorithm uses only full Nesterov-Todd (NT) step at each iteration to find an ε-approximated solution to SDO. Polynomial complexity of the proposed algorithm is established which is as good as the LO analogue. Finally, we present some numerical results to prove the efficiency of the proposed algorithm.

Sensor fault detection in a biohydrogen production process based on an interval observer and adaptive thresholds

In this paper, an interval observer-based fault detection strategy is proposed to deal with faults in the hydrogen flow rate sensor of a biohydrogen production process. A control Hinf-based robust interval observer is designed to (i) estimate the glucose and biomass concentrations from measurements of the hydrogen flow rate produced, (ii) attenuate the effect of the unknown inlet glucose concentration, and (iii) detect faults in the hydrogen flow rate sensor through adaptive thresholds. The gain is designed by solving a semi-definite optimization problem subject to LMIs. The feasibility of the proposed strategy is demonstrated by simulations.

Novel robust minimum error entropy wasserstein distribution kalman filter under model uncertainty and non-gaussian noise

Recently, the divergence-based minimax approach was proposed and introduced to robust Kalman filtering under model uncertainty and Gaussian noise, and it significantly outperformed the standard Kalman filter. In this study, we consider a new Wasserstein-distribution based Kalman filter in which model uncertainty and non-Gaussian noise exist simultaneously. First, we construct a joint Gaussian distribution of states and observations in state space. Then, we set the equivalent estimator joint Gaussian distribution as a Wasserstein ambiguity set allowable neighborhood centered on the true distribution. Thus, the estimator distribution problem is transformed into a minimax problem, and the model uncertainty is tolerated accordingly. Subsequently, we introduce minimum error entropy to optimize the minimax problem based on Wasserstein ambiguity sets, so as to handle the influence of non-Gaussian noise. The minimax problem constrained by the minimum error entropy is transformed into a semi-positive definite convex optimization problem. By constructing two iterative sub-problems that are mutually conditional, the nonlinear semi-definite program finite convex optimization problem is solved. Finally, the minimum error entropy Wasserstein distribution Kalman filter algorithm is proposed. Additionally, the convergence of the proposed algorithm is clarified, and its effectiveness verified by comparing a series of algorithms in typical simulation scenarios.

Implementation in Java of Algorithms for Semidefinite Optimization

Abstract We discuss the possibility of solving the semidefinite optimization problem using interior-point algorithms. We present the primal and dual semidefinite programming problems, and then determine the interior-point condition and the optimality criteria. We analyze the central path system and the modification of this, using the method of algebraically equivalent transformation. We use the Nesterov-Todd scaling technique to obtain the proper search directions. We give a modified version of the Nesterov-Todd step interior-point algorithm based on the implementation point of view. We present some numerical results based on a code implemented in the Java programming language. We compare the results obtained for the identity map and the square root function within the framework of the algebraically equivalent transformation technique.

Sparse Spatial Spectral Fitting with Nonuniform Noise Covariance Matrix Estimation Based on Semidefinite Optimization

In general, the azimuth estimation in array signal processing is derived under the assumption of uniform white noise, whose covariance matrix is a scaled identity matrix. However, in practice, the noise can be nonuniform with an arbitrary unknown diagonal covariance matrix. In this paper, the estimation of the noise covariance matrix is formulated into a solution to the semidefinite optimization problem which can obtain a more accurate sensor noise covariance matrix. In the proposed algorithm, the estimated nonuniform noise is subtracted from the sample covariance matrix. The simulation results show that the proposed algorithm can significantly improve the performance of the sparse spectrum fitting algorithm in nonuniform noise case, while the classic SpSF algorithm is used under uniform white noise assumption. The water pool experiments show that there are indeed significant differences in the noise covariance of the sensors.

A PRIMAL-DUAL IPMS FOR SDO PROBLEM BASED ON A NEW KERNEL FUNCTION WITH A LOGARITHMIC BARRIER TERM

In this paper, we consider primal-dual Interior Point Method $(IPMs)$ for semidefinite optimization problem $(SDO)$ problems, based on a new kernel function with a logarithmic barrier term, which play an important role for generating a new design of primaldual (IPM) algorithms. New search directions and proximity functions are proposed, based on this kernel function.We proved that our algorithm has $\mathbf{O}\left( qsn^{\frac{sq+1}{2sq}}\log \left( \frac{n}{\epsilon }\right) \right) $ iteration bound for large-update methods and $\mathbf{O}\left(q^{2}s^{2}\sqrt{n}\log \left( \frac{n}{\epsilon }\right) \right) $ iteration bound for small-update methods. Finally, for its numerical tests, some strategies are used and indicate that the algorithm is efficient.

On Computing the Nonlinearity Interval in Parametric Semidefinite Optimization

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painlevé–Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.

Experimental validation of an interval observer-based sensor fault detection strategy applied to a biohydrogen production dark fermenter

In this paper, the problem of designing an interval observer applied to a biohydrogen production dark fermenter is addressed, in order to: (i) estimate the glucose and biomass concentrations, and the hydrogen flow rate produced, (ii) reduce the influence of the unknown inlet glucose concentration and the model uncertainties, and (iii) detect the presence of faults in the hydrogen flow rate sensor. Based on literature reports, we propose an interval observer structure, constituted by a Luenberger observer and an interval predictor, for a class of linear systems. The interval observer offers three estimations: the upper and lower estimations provided by the interval predictor, and one estimation given by the Luenberger observer. The upper and lower estimations provide bounds at any instant of time for the state vector of the perturbed/uncertain biohydrogen production bioreactor model from an appropriate initial condition. The design conditions presented in this work are based on control H∞ in combination with pole placement and polytopic parametric uncertainties. The observer design is proposed as a semi-definite optimization problem subject to Linear Matrix Inequalities, which do not depend on the cooperativity property. Based on the proposed observer structure, we also present two fault detection schemes for the same class of uncertain systems to detect the occurrence of sensor faults: one makes use of the logic comparison between the set threshold and the residual produced by the Luenberger observer; the other considers adaptive thresholds for the output signal by using the upper and lower estimates of the interval predictor. Finally, the proposed approaches are validated with experimental data from a biohydrogen production dark fermenter.

A Modified Method for Beam-Forming Using Covariance Matrix in MIMO Radar System

A MIMO radar provides solution for beamforming, since transmit beampattern synthesis is well proven method for stimulating the antenna array to develop the beampattern. It is close to the desired one which can give minimum error between these two. A covariance matrix in beamforming contains the direction of invariable data and magnitude distribution in multidimensional space which decides the closeness with the desired beampattern. Here, selection of an optimal covariance matrix is constrained optimization problem with minimization of cost function. In high directivity radar systems, the antenna beam is needed to be steered to cover a large area for detection. In this proposed work, minimization of convex function and transmit beampattern synthesis is carried out with modifications of the covariance matrix ‘Rcov’ for minimum and maximum transmission power utilization constraints. The beampattern design problem is reformulated here, as semi-definite programming optimization problem and solved using convex optimization, with MATLAB simulation platform. The results show that the beampattern generated using this modified covariance matrix is optimally close to desired pattern.