On representations of the feasible set in convex optimization

We consider the convex optimization problem $\min \{f(x) : g_j(x)\leq 0, j=1,...,m\}$ where $f$ is convex, the feasible set K is convex and Slater's condition holds, but the functions $g_j$ are not necessarily convex. We show that for any representation of K that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the $g_j$ are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of K and not so much its representation.Comment: to appear in Optimization Letter

Counting and computing regions of DD-decomposition: algebro-geometric approach

New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying $D$-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure

Pose-informed deep learning method for SAR ATR

Synthetic aperture radar (SAR) images for automatic target classification (automatic target recognition (ATR)) have attracted significant interest as they can be acquired day and night under a wide range of weather conditions. However, SAR images can be time consuming to analyse, even for experts. ATR can alleviate this burden and deep learning is an attractive solution. A new deep learning Pose-informed architecture solution, that takes into account the impact of target orientation on the SAR image as the scatterers configuration changes, is proposed. The classification is achieved in two stages. First, the orientation of the target is determined using a Hough transform and a convolutional neural network (CNN). Then, classification is achieved with a CNN specifically trained on targets with similar orientations to the target under test. The networks are trained with translation and SAR-specific data augmentation. The proposed Pose-informed deep network architecture was successfully tested on the Military Ground Target Dataset (MGTD) and the Moving and Stationary Target Acquisition and Recognition (MSTAR) datasets. Results show the proposed solution outperformed standard AlexNets on the MGTD, MSTAR extended operating condition (EOC)1, EOC2 and standard operating condition (SOC)10 datasets with a score of 99.13% on the MSTAR SOC10