Semidefinite Optimization Problems Research Articles

Interior-point methods based on kernel functions for symmetric optimization

We present a generalization to symmetric optimization of interior-point methods for linear optimization based on kernel functions. Symmetric optimization covers the three most common conic optimization problems: linear, second-order cone and semi-definite optimization problems. Namely, we adapt the interior-point algorithm described in Peng et al. [Self-regularity: A New Paradigm for Primal–Dual Interior-point Algorithms. Princeton University Press, Princeton, NJ, 2002.] for linear optimization to symmetric optimization. The analysis is performed through Euclidean Jordan algebraic tools and a complexity bound is derived.

On the Generalized Gaussian CEO Problem

This paper considers a distributed source coding (DSC) problem where L encoders observe noisy linear combinations of K correlated remote Gaussian sources, and separately transmit the compressed observations to the decoder to reconstruct the remote sources subject to a sum-distortion constraint. This DSC problem is referred to as the generalized Gaussian CEO problem since it can be viewed as a generalization of the quadratic Gaussian CEO problem where the number of remote source K=1. First, we provide a new outer region obtained using the entropy power inequality and an equivalent argument (in the sense of having the same rate-distortion region and Berger-Tung inner region) among a certain class of generalized Gaussian CEO problems. We then give two sufficient conditions for our new outer region to match the inner region achieved by Berger-Tung schemes, where the second matching condition implies that in the low-distortion regime, the Berger-Tung inner rate region is always tight, while in the high-distortion regime, the same region is tight if a certain condition holds. The sum-rate part of the outer region is also studied and shown to meet the Berger-Tung sum-rate upper bound under a certain condition, which is obtained using the Karush-Kuhn-Tucker conditions of the underlying convex semidefinite optimization problem, and is in general weaker than the aforesaid two for rate region tightness.

Sufficient conditions for global optimality of semidefinite optimization

In this article, by using the Lagrangian function, we investigate the sufficient global optimality conditions for a class of semi-definite optimization problems, where the objective function are general nonlinear, the variables are mixed integers subject to linear matrix inequalities (LMIs) constraints as well as bounded constraints. In addition, the sufficient global optimality conditions for general nonlinear programming problems are derived, where the variables satisfy LMIs constraints and box constraints or bivalent constraints. Furthermore, we give the sufficient global optimality conditions for standard semi-definite programming problem, where the objective function is linear, the variables satisfy linear inequalities constraints and box constraints. Mathematics Subject Classification 2010: 90C30; 90C26; 90C11.

Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term

In this paper we propose primal-dual interior-point algorithms for semidefinite optimization problems based on a new kernel function with a trigonometric barrier term. We show that the iteration bounds are \(O(\sqrt{n}\log(\frac{n}{\epsilon}))\) for small-update methods and \(O(n^{\frac{3}{4}}\log(\frac{n}{\epsilon}))\) for large-update, respectively. The resulting bound is better than the classical kernel function. For small-update, the iteration complexity is the best known bound for such methods.

On a measure of distance for quantum strategies

The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Gutoski and Watrous [In Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science (STACS’05), Lecture Notes in Computer Science, Vol. 3404 (Springer, 2005), pp. 605–616. The generalized result states that for any two convex sets S0, S1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of S0 ∈ S0 from any choice of S1 ∈ S1 with bias proportional to the minimal distance between the sets S0 and S1 as measured by this norm. A similar discrimination result for channels then follows as a special case.

Notes on Computational Complexity of GE Inequalities

This paper is a revision of my paper, CFDP 1865. The principal innovation is an equivalent reformulation of the decision problem for weak feasibility of the GE inequalities, using polynomial time ellipsoid methods, as a semidefinite optimization problem, using polynomial time interior point methods. We minimize the maximum of the Euclidean distances between the aggregate endowment and the Minkowski sum of the sets of consumer's Marshallian demands in each observation. We show that this is an instance of the generic semidefinite optimization problem: inf_{x in K}f(x) equivalent to Opt(K,f), the optimal value of the program, where the convex feasible set K and the convex objective function f(x) have semidefinite representations. This problem can be approximately solved in polynomial time. That is, if p(K,x) is a convex measure of infeasibilty, where for all x, p(K,x) >= 0 and p(K,z) = 0 iff z in K, then for every epsilon > 0 there exists an epsilon-optimal y such that p(K,y)

On the Performance of Optimal Input Signals for Frequency Response Estimation

We consider the problem of minimum-variance excitation design for frequency response estimation based on finite impulse response (FIR) and output error (OE) models. The objective is to minimize the power of the input signal to be used in the system identification experiment subject to a model accuracy constraint. For FIR and OE models this leads to a finite dimensional semi-definite programming optimization problem. We study, in detail, how to apply this approach to the estimation of the frequency response at a given frequency, . The first case concerns minimizing the asymptotic variance of the estimated frequency response based on an FIR model estimate. We compare the optimal input signal with a sinusoidal signal with frequency that gives the same model accuracy, and show that the input power can, at best, be reduced by a factor of two when using the optimal input signal. Conditions are given under which the sinusoidal signal is optimal, and it is shown that this is a common case for higher order FIR models. Next, we study FIR model based estimation of the absolute value and phase of the frequency response at a given frequency, . We derive the corresponding optimal input signals and compare their performances with that of a sinusoidal input signal with frequency . The relative reduction of input power when using the optimal solution is at best a factor of two. Finally, we discuss how to extend the FIR results to OE system identification by using an input parametrization proposed by Stoica and Soderstrom (1982).

An inexact spectral bundle method for convex quadratic semidefinite programming

We present an inexact spectral bundle method for solving convex quadratic semidefinite optimization problems. This method is a first-order method, hence requires much less computational cost in each iteration than second-order approaches such as interior-point methods. In each iteration of our method, we solve an eigenvalue minimization problem inexactly, and solve a small convex quadratic semidefinite program as a subproblem. We give a proof of the global convergence of this method using techniques from the analysis of the standard bundle method, and provide a global error bound under a Slater type condition for the problem in question. Numerical experiments with matrices of order up to 3000 are performed, and the computational results establish the effectiveness of this method.

Simplified infeasible interior-point algorithm for SDO using full Nesterov-Todd step

Interior-point methods for semidefinite optimization problems have been studied frequently, due to their polynomial complexity and practical implications. In this paper we propose a primal-dual infeasible interior-point algorithm that uses full Nesterov-Todd (NT) steps with a different feasibility step. We obtain the currently best known iteration bound for semidefinite optimization problems.

Learning Linear and Nonlinear PCA with Linear Programming

An SVM-like framework provides a novel way to learn linear principal component analysis (PCA). Actually it is a weighted PCA and leads to a semi-definite optimization problem (SDP). In this paper, we learn linear and nonlinear PCA with linear programming problems, which are easy to be solved and can obtain the unique global solution. Moreover, two algorithms for learning linear and nonlinear PCA are constructed, and all principal components can be obtained. To verify the performance of the proposed method, a series of experiments on artificial datasets and UCI benchmark datasets are accomplished. Simulation results demonstrate that the proposed method can compete with or outperform the standard PCA and kernel PCA (KPCA) in generalization ability but with much less memory and time consuming.

On methods for solving nonlinear semidefinite optimization problems

The nonlinear semidefinite optimization problem arises from applications in system control, structural design, financial management, and other fields. However, much work is yet to be done to effectively solve this problem. We introduce some new theoretical and algorithmic development in this field. In particular, we discuss first and second-order algorithms that appear to be promising, which include the alternating direction method, the augmented Lagrangian method, and the smoothing Newton method. Convergence theorems are presented and preliminary numerical results are reported.

A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs

We propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems. The method is a first-order method, therefore requires much less computational effort per iteration than the second-order approaches such as the interior point methods or the smoothing Newton methods. In fact, only a single inexact metric projection onto the positive semidefinite cone is required at each iteration. We prove global convergence and provide numerical evidence to show the effectiveness of this method.

Convex Inequalities Without Constraint Qualification nor Closedness Condition, and Their Applications in Optimization

Given two convex lower semicontinuous extended real valued functions F and h defined on locally convex spaces, we provide a dual transcription of the relation $$ F\left( 0,\cdot \right) \geq h\left( \cdot \right) . $$ (⋆) Some results in this direction are obtained in the first part of the paper (Lemma 2, Theorem 1). These results then are applied to the case when the left-hand-side in (⋆) is the sum of two convex functions with a convex composite one (Theorem 2). In the spirit of previous works (Hiriart-Urruty and Phelps, J Funct Anal 118:154–166, 1993; Penot, J Convex Anal 3:207–219, 1996, 2005; Thibault, 1995, SIAM J Control Optim 35:1434–1444, 1997, etc.) we give in Theorem 3 a formula for the subdifferential of such a function without any qualification condition. As a consequence of that, we extend to the nonreflexive setting a recent result (Jeyakumar et al., J Glob Optim 36:127–137 2006, Theorem 3.2) about subgradient optimality conditions without constraint qualifications. Finally, we apply Theorem 2 to obtain Farkas-type lemmas and new results on DC, convex, semi-definite, and linear optimization problems.

Newsvendor games: convex optimization of centralized inventory operations

A finite set of outlets with randomly fluctuating demands bands together to reduce costs by buying, storing and distributing their inventory jointly. This is termed inventory centralization and is a type of risk pooling. The expected centralization cost can be lowered even further, without disrupting the demand behavior at individual outlets, by inducing the outlets to correlate their individual demands. Given that the outlets’ demands are normally distributed, the lowering of the centralized cost corresponds to a semidefinite optimization problem. This paper establishes a closed-form optimal solution of the semidefinite program and a fair allocation of the centralized cost at optimality.

Compact Modeling of Nonlinear Analog Circuits Using System Identification via Semidefinite Programming and Incremental Stability Certification

This paper presents a system identification technique for generating stable compact models of typical analog circuit blocks in radio frequency systems. The identification procedure is based on minimizing the model error over a given training data set subject to an incremental stability constraint, which is formulated as a semidefinite optimization problem. Numerical results are presented for several analog circuits, including a distributed power amplifier, as well as a MEM device. It is also shown that our dynamical models can accurately predict important circuit performance metrics, and may thus, be useful for design optimization of analog systems.

A Polynomial-time Interior-point Algorithm for Convex Quadratic Semidefinite Optimization

In this paper we propose a primal-dual interior-point algorithm for convex quadratic semidefinite optimization problem. The search direction of algorithm is defined in terms of a matrix function and the iteration is generated by full-Newton step. Furthermore, we derive the iteration bound for the algorithm with small-update method, namely, O( $\sqrt{n}$ log $\frac{n}{\varepsilon}$ ), which is best-known bound so far.

ϵ-Duality Theorems for Convex Semidefinite Optimization Problems with Conic Constraints

A convex semidefinite optimization problem with a conic constraint is considered. We formulate a Wolfe-type dual problem for the problem for its -approximate solutions, and then we prove -weak duality theorem and -strong duality theorem which hold between the problem and its Wolfe type dual problem. Moreover, we give an example illustrating the duality theorems.

A New Class of Large Neighborhood Path-Following Interior Point Algorithms for Semidefinite Optimization with $O(\sqrt{n}\log\frac{\mathrm{Tr}(X^0S^0)}{\epsilon})$ Iteration Complexity

In this paper, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood $\mathcal{N}(\tau_1,\tau_2,\eta)$, and, as usual but with a small change, we make use of the scaled Newton equations for symmetric search directions. After defining the “positive part” and the “negative part” of a symmetric matrix, we recommend solving the Newton equation with its right-hand side replaced first by its positive part and then by its negative part, respectively. In such a way, we obtain a decomposition of the classical Newton direction and use different step lengths for each of them. Starting with a feasible point $(X^0,y^0,S^0)$ in $\mathcal{N}(\tau_1,\tau_2,\eta)$, the algorithm terminates in at most $O(\eta\sqrt{\kappa_{\infty}n}\log({Tr}(X^0S^0)/\epsilon))$ iterations, where $\kappa_{\infty}$ is a parameter associated with the scaling matrix $P$ and $\epsilon$ is the required precision. To our best knowledge, when the parameter $\eta$ is a constant, this is the first large neighborhood path-following interior point method (IPM) with the same complexity as small neighborhood path-following IPMs for semidefinite optimization that use the Nesterov-Todd direction. In the case where $\eta$ is chosen to be in the order of $\sqrt{n}$, our result coincides with the results for classical large neighborhood IPMs. Some preliminary numerical results also confirm the efficiency of the algorithm.

General solutions for information-based and Bayesian approaches to model selection in linear regression and their equivalence

In the paper we consider the problem of model selection for linear regression within Bayesian and information-based frameworks. For both cases we generalize known approaches (evidence-based and Akaike information criterion) and derive criterion functions in terms of (in general case non-factorial) weight priors which are assumed to be Gaussian. Optimization of these criterion functions leads to two semidefinite optimization problems which can be solved analytically. We present a method that finds best priors in both approaches and show their equivalence. Surprisingly it appears that optimal prior has rank one covariance matrix. We derive explicit condition of degenerative decision rule, i.e., regression with all weights equal to zero. We conclude with experiments that show that the proposed approach significantly reduces the time needed for model selection in comparison with alternatives based on cross-validation and iterative evidence maximization while keeping generalization ability

Transmission Strategies in MIMO Ad Hoc Networks

Precoding problem in multiple-input multiple-output (MIMO) ad hoc networks is addressed in this work. Firstly, we consider the problem of maximizing the system mutual information under a power constraint. In this context, we give a brief overview of the nonlinear optimization methods, and systematically we compare their performances. Then, we propose a fast and distributed algorithm based on the quasi-Newton methods to give a lower bound of the system capacity of MIMO ad hoc networks. Our proposed algorithm solves the maximization problem while diminishing the amount of information in the feedback links needed in the cooperative optimization. Secondly, we propose a different problem formulation, which consists in minimizing the total transmit power under a quality of signal constraint. This novel problem design is motivated since the packets are captured in ad hoc networks based on their signal-to-interference-plus-noise ratio (SINR) values. We convert the proposed formulation into semidefinite optimization problem, which can be solved numerically using interior point methods. Finally, an extensive set of simulations validates the proposed algorithms.