Semidefinite Programming Techniques Research Articles

A robust geostatistical approach to contaminant source identification

Estimation under model uncertainty remains a practical concern in many scientific and engineering fields. A commonly encountered example in groundwater remediation is the contaminant source identification problem. Like many other inverse problems, contaminant source identification is inherently ill posed and is sensitive to both data and model uncertainties. Model uncertainties, which may be introduced at virtually any stage of a model building process, can adversely affect estimator performance if they are not accounted for properly. In this paper, a robust geostatistical approach (RGS), which is extended from the geostatistical inversion approach (GS), is used to solve linear estimation problems. The uncertainties in both model and covariance matrices are taken into account in the RGS formulation. The nominal correlation structural parameters are estimated using a structural analysis procedure. The resulting minimax optimization problem is solved using semidefinite programming techniques. The RGS is generic and can be applied to any problem for which the GS is suitable and the upper bound of uncertainty can be quantified. The RGS is illustrated for source release history identification in a two‐dimensional aquifer where the model uncertainty is caused by variability in hydraulic conductivity. It is shown that when the model is perfectly known, the RGS solution coincides with that of the GS; when the model has uncertainty, the RGS is robust against unknown variations from a nominal model, and its overall performance is better than that of the GS using the same nominal model.

LMI Approximations for Cones of Positive Semidefinite Forms

An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms (homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.

Learning the Kernel Matrix with Semidefinite Programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space---classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm---using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

Upper bounds and exact algorithms for [formula omitted]-dispersion problems

The p-dispersion-sum problem is the problem of locating p facilities at some of n predefined locations, such that the distance sum between the p facilities is maximized. The problem has applications in telecommunication (where it is desirable to disperse the transceivers in order to minimize interference problems), and in location of shops and service-stations (where the mutual competition should be minimized). A number of fast upper bounds are presented based on Lagrangian relaxation, semidefinite programming and reformulation techniques. A branch-and-bound algorithm is then derived, which at each branching node is able to compute the reformulation-based upper bound in O ( n ) time. Computational experiments show that the algorithm may solve geometric problems of size up to n = 90 , and weighted geometric problems of size n = 250 . The related p-dispersion problem is the problem of locating p facilities such that the minimum distance between two facilities is as large as possible. New formulations and fast upper bounds are presented, and it is discussed whether a similar framework as for the p-dispersion sum problem can be used to tighten the upper bounds. A solution algorithm based on transformation of the p-dispersion problem to the p-dispersion-sum problem is finally presented, and its performance is evaluated through several computational experiments.

Learning recurrent neural models with minimal complexity from neural tuning data

A learning algorithm for the estimation of the structure of nonlinear recurrent neural models from neural tuning data is presented. The proposed method combines support vector regression with additional constraints that result from a stability analysis of the dynamics of the fitted network model. The optimal solution can be determined from a single convex optimization problem that can be solved with semidefinite programming techniques. The method successfully estimates the feed-forward and the recurrent connectivity structure of neural field models using as data only examples of stable stationary solutions of the neural dynamics. The class of neural models that can be learned is quite general. The only a priori assumptions are the translation invariance and the smoothness of the feed-forward and recurrent spatial connectivity profile. The efficiency of the method is illustrated by comparing it with estimates based on radial basis function networks and support vector regression.

On Convex Quadratic Approximation

In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of statistics and optimization. We show that convexity can be enforced in the multivariate case by using semidefinite programming techniques. Key words: Convex function, least squares, quadratic interpolation, semidefinite programming

On convex quadratic approximation

In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of statistics and optimization. We show that convexity can be enforced in the multivariate case by using semidefinite programming techniques.

Robust Filtering via Semidefinite Programming with Applications to Target Tracking

In this paper we propose a novel finite-horizon, discrete-time, time-varying filtering method based on the robust semidefinite programming (SDP) technique. The proposed method provides robust performance in the presence of norm-bounded parameter uncertainties in the system model. The robust performance of the proposed method is achieved by minimizing an upper bound on the worst-case varianceof the estimation error for all admissible systems. Our method is recursive and computationally efficient. In our simulations, the new method provides superior performance to some of the existing robust filtering approaches. In particular, when applied to the problem of target tracking, the new method has led to a significant improvement in tracking performance. Our work shows that the robust SDP technique and the interior point algorithms can bring substantial benefits to practically important engineering problems.

Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colorability

Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAX-CUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of $$\frac{1}{2} + \varepsilon$$ . This is an improvement over the ratio of $$\frac{1}{2}$$ attributed to another local improvement heuristic for MAX-CUT (C. Papadimitriou, Computational Complexity, Addison Wesley, 1994). In fact we provide a bound of $$\frac{k}{{k + 1}}$$ for this problem, where k ≥ 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques (M. Goemans and D.P. Williamson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994a, pp. 422–431) appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both (M. Goemans and D.P. Williamson, SIAM J. Disc. Math, vol. 7, pp. 656–666, 1994b) yield a bound of $$\frac{1}{2}$$ for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of $$\frac{1}{2} + \varepsilon$$ for this problem.

Semidefinite Programming Techniques for Reduced Order Systems with Guaranteed Stability Margins

In this paper, the compensator based reduced order control design framework of Burns and King (J. Vibrations and Control, vol. 4, pp. 297–323, 1998) is modified to yield low order systems with guaranteed stability margins. This result is achieved through use of a logarithmic barrier function. In addition, a reduced basis method is formulated in which the compensator equations are approximated on uneven gridss guaranteed stability margins are also included. The methods are tested numerically on a one dimensional, nonlinear, damped, hyperbolic structural control problem. Examples are provided to illustrate differences between systems designed by both reduced basis methods.

Structured approach to storage allocation for improved process controllability

AbstractAn optimization‐based approach is presented that determines how plant feedstocks should be allocated to storage when there are fewer storage vessels than feedstocks. It is assumed here that material from the storage vessels will be subsequently blended for processing in downstream processes. The objective of the feedstock allocation strategy is chosen to ensure maximum flexibility for downstream process operation. Given the stated objective for feedstock allocation and the physical constraints, the feedstock storage allocation problem is posed in optimization form. The solution of the resulting singular‐value optimization problem is discussed in terms of semidefinite programming techniques. The ideas presented are illustrated using a crude‐oil storage case study. Finally, a number of observations regarding useful extehsions to the proposed methods are presented.