Related papers: Convex Duality Made Difficult
This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We…
Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data,…
Non-convex optimization problems have multiple local optimal solutions. Non-convex optimization problems are commonly found in numerous applications. One of the methods recently proposed to efficiently explore multiple local optimal…
Despite the recent development in machine learning, most learning systems are still under the concept of "black box", where the performance cannot be understood and derived. With the rise of safety and privacy concerns in public, designing…
The convergence rate of various first-order optimization algorithms is a pivotal concern within the numerical optimization community, as it directly reflects the efficiency of these algorithms across different optimization problems. Our…
Functional constrained optimization is becoming more and more important in machine learning and operations research. Such problems have potential applications in risk-averse machine learning, semisupervised learning, and robust optimization…
The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these…
This material provides thorough tutorials on some optimization techniques frequently used in various engineering disciplines, including convex optimization, linearization techniques and mixed-integer linear programming, robust optimization,…
We focus on establishing the foundational paradigm of a novel optimization theory based on convolution with convex kernels. Our goal is to devise a morally deterministic model of locating the global optima of an arbitrary function, which is…
This book is devoted to finite-dimensional problems of non-convex non-smooth optimization and numerical methods for their solution. The problem of nonconvexity is studied in the book on two main models of nonconvex dependencies: these are…
Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
A new and simple method for quasi-convex optimization is introduced from which its various applications can be derived. Especially, a global optimum under constrains can be approximated for all continuous functions.
Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results…
Proximal operators with affine constraints arise in numerous models in nonconvex projection, composite optimization, and structured regularization. However, their efficient computation remains challenging due to the simultaneous presence of…
Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of the well-known convexity/concavity properties in the continuous domain. Submodular functions exhibit strong structure that lead to…
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show…
To every nearly convex optimization problem, that is a minimization problem with a nearly convex objective function and a nearly convex constraint set, we associate a uniquely defined convex optimization problem with a lower semicontinuous…
The paper introduces several new concepts for solving nonconvex or nonsmooth optimization problems, including convertible nonconvex function, exact convertible nonconvex function and differentiable convertible nonconvex function. It is…
In this paper we associate with an infinite family of real extended functions defined on a locally convex space, a sum, called robust sum, which is always well-defined. We also associate with that family of functions a dual pair of problems…