Compound optimal design of experiments – Semidefinite Programming formulations

We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in Ohsaki et al. (Comp. Meth. Appl. Mech. Eng. 180:203–217, 1999). We show how one may automatically obtain symmetric designs, by eliminating the ‘redundant’ symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.

Semidefinite programming (SDP) solvers are increasingly used as primitives in many program verification tasks to synthesize and verify polynomial invariants for a variety of systems including programs, hybrid systems and stochastic models. On one hand, they provide a tractable alternative to reasoning about semi-algebraic constraints. However, the results are often unreliable due to “numerical issues” that include a large number of reasons such as floating-point errors, ill-conditioned problems, failure of strict feasibility, and more generally, the specifics of the algorithms used to solve SDPs. These issues influence whether the final numerical results are trustworthy or not. In this paper, we briefly survey the emerging use of SDP solvers in the static analysis community. We report on the perils of using SDP solvers for common invariant synthesis tasks, characterizing the common failures that can lead to unreliable answers. Next, we demonstrate existing tools for guaranteed semidefinite programming that often prove inadequate to our needs. Finally, we present a solution for verified semidefinite programming that can be used to check the reliability of the solution output by the solver and a padding procedure that can check the presence of a feasible nearby solution to the one output by the solver. We report on some successful preliminary experiments involving our padding procedure.

We present a unified framework for analyzing the convergence of distributed optimization algorithms by formulating a semidefinite program (SDP) which can be efficiently solved to bound the linear rate of convergence. Two different SDP formulations are considered. First, we formulate an SDP that depends explicitly on the gossip matrix of the network graph. This result provides bounds that depend explicitly on the graph topology, but the SDP dimension scales with the size of the graph. Second, we formulate an SDP that depends implicitly on the gossip matrix via its spectral gap. This result provides coarser bounds, but yields a small SDP that is independent of graph size. Our approach improves upon existing bounds for the algorithms we analyzed, and numerical simulations reveal that our bounds are likely tight. The efficient and automated nature of our analysis makes it a powerful tool for algorithm selection and tuning, and for the discovery of new algorithms as well.

The sensor network localization (SNL) problem in embedding dimension r consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming (SDP) completion problem by using the linear mapping between Euclidean distance matrices (EDM) and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem.

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set \(K\) described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set \(K\) is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that \(K\) has a strictly feasible point.

In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.

In contrast to many other convex optimization classes, state-of-the-art semidefinite programming solvers are still unable to efficiently solve large-scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The key characteristic of the proposed algorithm is to be able to exploit the low-rank property inherent to several semidefinite programming problems. Exploiting the low-rank structure provides a substantial speedup and allows the operator splitting method to efficiently scale to larger instances. As opposed to other low-rank based methods, the proposed algorithm has convergence guarantees for general semidefinite programming problems. Additionally, an open-source semidefinite programming solver called ProxSDP is made available and its implementation details are discussed. Case studies are presented in order to evaluate the performance of the proposed methodology.

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.

For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.

Semi-supervised kernel learning methods have been received much more attention in the past few years. Traditional semi-supervised Non-Parametric Kernel Learning (NPKL) methods usually formulate the learning task as a Semi-Definite Programming (SDP) problem, which is very time consuming. Although some fast semi-supervised NPKL methods have been proposed recently, they usually scale very poorly. Furthermore, many semi-supervised NPKL methods are developed based on the manifold assumption. But, such an assumption might be invalid when handling some high-dimensional and sparse data, which has severely negative effect on the performance of learning algorithms. In this paper, we propose a more efficient semi-supervised NPKL method, which can effectively learn a low-rank kernel matrix from must-link and cannot-link constraints. Specially, by virtue of the nonlinear spectral embedded technique based on extreme learning machine (ELM), the proposed method has the ability of coping with data points that do not have a clear manifold structure in a low dimensional space. The proposed method is formulated as a trace ratio optimization problem, which is combined with dimensionality reduction in ELM feature space and aims to find optimal low-rank kernel matrices. The proposed optimization problem can be solved much more efficiently than SDP solvers. Extensive experiments have validated the superior performance of the proposed method compared to state-of-the-art semi-supervised kernel learning methods.

A gridless method for direction finding with sparse arrays in nonuniform noise

The quantification of heating, ventilation, and air condition (HVAC) system flexibility is critical to the operations of both the grid and buildings in demand response (DR) programs. However, the flexibility quantification is challenging due to the non-linearity and non-convexity of thermal dynamics associated with HVAC components. This paper proposes a novel HVAC flexibility quantification method based on a semidefinite programming (SDP) formulation. The SDP is reformulated from the non-convex problem of HVAC power optimization, and can be solved efficiently in real-time. The physics-based HVAC model is incorporated to ensure the reliability and accuracy of solutions. The quantification results are organized into an HVAC flexibility table that can provide response strategies on adjusting HVAC setpoints in response to the grid signals received. The developed response strategies minimize occupant discomfort while satisfying grid requirements. A case study of a test building model is carried out to illustrate the flexibility quantification framework and compares the performance of two DR strategies.

Semidefinite programming solution of economic dispatch problem with non-smooth, non-convex cost functions

Contemporary practical methods for protein nuclear magnetic resonance (NMR) structure determination use molecular dynamics coupled with a simulated annealing schedule. The objective of these methods is to minimize the error of deviating from the nuclear overhauser effect (NOE) distance constraints. However, the corresponding objective function is highly nonconvex and, consequently, difficult to optimize. Euclidean distance matrix (EDM) methods based on semidefinite programming (SDP) provide a natural framework for these problems. However, the high complexity of SDP solvers and the often noisy distance constraints provide major challenges to this approach. The main contribution of this article is a new SDP formulation for the EDM approach that overcomes these two difficulties. We model the protein as a set of intersecting two- and three-dimensional cliques. Then, we adapt and extend a technique called semidefinite facial reduction to reduce the SDP problem size to approximately one quarter of the size of the original problem. The reduced SDP problem can be solved approximately 100 times faster, and it is also more resistant to numerical problems from erroneous and inexact distance bounds.

The complexity-performance trade-off is a fundamental aspect of the design of low-density parity-check (LDPC) codes. In this paper, we consider LDPC codes for the binary erasure channel (BEC), use code rate for performance metric, and number of decoding iterations to achieve a certain residual erasure probability for complexity metric. We first propose a quite accurate approximation of the number of iterations for the BEC. Moreover, a simple but efficient utility function corresponding to the number of iterations is developed. Using the aforementioned approximation and the utility function, two optimization problems w.r.t. complexity are formulated to find the code degree distributions. We show that both optimization problems are convex. In particular, the problem with the proposed approximation belongs to the class of semi-infinite problems which are computationally challenging to be solved. However, the problem with the proposed utility function falls into the class of semi-definite programming (SDP) and thus, the global solution can be found efficiently using available SDP solvers. Numerical results reveal the superiority of the proposed code design compared to existing code designs from literature.