Time-Varying Semidefinite Programs

We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data vary polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value of the TV-SDP. Moreover, by using a Positivstellensatz (positive locus theorem) on univariate polynomial matrices, we show that the best polynomial solution of a given degree to a TV-SDP can be found by solving a semidefinite program of tractable size. We also provide a sequence of dual problems that can be cast as SDPs and that give upper bounds on the optimal value of a TV-SDP (in maximization form). We prove that under a boundedness assumption, this sequence of upper bounds converges to the optimal value of the TV-SDP. Under the same assumption, we also show that the optimal value of the TV-SDP is attained. We demonstrate the efficacy of our algorithms on a maximum-flow problem with time-varying edge capacities, a wireless coverage problem with time-varying coverage requirements, and on biobjective semidefinite optimization where the goal is to approximate the Pareto curve in one shot.

Three candidate plurality is stablest for small correlations

Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$ -dimensional, if $m-1\leq n$ . In particular, the maximum noise stability of a partition of m sets in $\mathbb {R}^{n}$ of fixed Gaussian volumes is constant for all n satisfying $n\geq m-1$ . From this result, we obtain: (i) A proof of the plurality is stablest conjecture for three candidate elections, for all correlation parameters $\rho $ satisfying $0<\rho <\rho _{0}$ , where $\rho _{0}>0$ is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell’s inequality (corresponding to the case $m=2$ ). The structure theorem answers a question of De–Mossel–Neeman and of Ghazi–Kamath–Raghavendra. Item (i) is the first proof of any case of the plurality is stablest conjecture of Khot-Kindler-Mossel-O’Donnell for fixed $\rho $ , with the case $\rho \to L1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze–Jerrum semidefinite program for solving MAX-3-CUT, assuming the unique games conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the plurality is stablest conjecture is known to be false.

Distribution-Free Inventory Risk Pooling in a Multilocation Newsvendor

We study a multilocation newsvendor network when the only information available on the joint distribution of demands are the values of its mean vector and covariance matrix. We adopt a distributionally robust model to find inventory levels that minimize the worst-case expected cost among the distributions consistent with this information. This problem is NP-hard. We find a closed-form tight bound on the expected cost when there are only two locations. This bound is tight under a family of joint demand distributions with six support points. For the general case, we develop a computationally tractable upper bound on the worst-case expected cost if the costs of fulfilling demands have a nested structure. This upper bound is the optimal value of a semidefinite program whose dimensions are polynomial in the number of locations. We propose an algorithm that can approximate general fulfillment cost structures by nested structures, yielding a computationally tractable heuristic for distributionally robust inventory optimization on general newsvendor networks. We conduct experiments on networks resembling U.S. e-commerce distribution networks to show the value of a distributionally robust approach over a stochastic approach that assumes an incorrect demand distribution. This paper was accepted by Chung Piaw Teo, optimization.

Sampling the feasible sets of SDPs and volume approximation

We present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to dimension 200. We illustrate its efficiency on various data sets.

Optimal probes and error-correction schemes in multi-parameter quantum metrology

We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.

Reweighted Covariance Fitting Based on Nonconvex Schatten-p Minimization for Gridless Direction of Arrival Estimation

In this paper, we reformulate the gridless direction of arrival (DoA) estimation problem in a novel reweighted covariance fitting (CF) method. The proposed method promotes joint sparsity among different snapshots by means of nonconvex Schatten-p quasi-norm penalty. Furthermore, for more tractable and scalable optimization problem, we apply the unified surrogate for Schatten-p quasi-norm with two-factor matrix norms. Then, a locally convergent iterative reweighted minimization method is derived and solved efficiently via a semidefinite program using the optimization toolbox. Finally, numerical simulations are carried out in the background of unknown nonuniform noise and under the consideration of coprime array (CPA) structure. The results illustrate the superiority of the proposed method in terms of resolution, robustness against nonuniform noise, and correlations of sources, in addition to its applicability in a limited number of snapshots.

Solving the Time- and Frequency-Multiplexed Problem of Constrained Radiofrequency Induced Hyperthermia

Targeted radiofrequency (RF) heating induced hyperthermia has a wide range of applications, ranging from adjunct anti-cancer treatment to localized release of drugs. Focal RF heating is usually approached using time-consuming nonconvex optimization procedures or approximations, which significantly hampers its application. To address this limitation, this work presents an algorithm that recasts the problem as a semidefinite program and quickly solves it to global optimality, even for very large (human voxel) models. The target region and a desired RF power deposition pattern as well as constraints can be freely defined on a voxel level, and the optimum application RF frequencies and time-multiplexed RF excitations are automatically determined. 2D and 3D example applications conducted for test objects containing pure water (rtarget = 19 mm, frequency range: 500–2000 MHz) and for human brain models including brain tumors of various size (r1 = 20 mm, r2 = 30 mm, frequency range 100–1000 MHz) and locations (center, off-center, disjoint) demonstrate the applicability and capabilities of the proposed approach. Due to its high performance, the algorithm can solve typical clinical problems in a few seconds, making the presented approach ideally suited for interactive hyperthermia treatment planning, thermal dose and safety management, and the design, rapid evaluation, and comparison of RF applicator configurations.

A Note on the Hierarchy of Quantum Measurement Incompatibilities

The quantum measurement incompatibility is a distinctive feature of quantum mechanics. We investigate the incompatibility of a set of general measurements and classify the incompatibility by the hierarchy of compatibilities of its subsets. By using the approach of adding noises to measurement operators, we present a complete classification of the incompatibility of a given measurement assemblage with n members. Detailed examples are given for the incompatibility of unbiased qubit measurements based on a semidefinite program.