In this paper, we consider a framework of inexact proximal point methods for convex optimization that allows a relative error tolerance in the approximate solution of each proximal subproblem and establish its convergence rate. We then show that the well-known forward-backward splitting algorithm for convex optimization belongs to this framework. Finally, we propose and establish the iteration-...

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization ...

Convex optimization has the advantages on solving the problems with hundreds of variables and thousands of constraints. However, traditional multi-body model based zero moment point (ZMP) formulation makes it impossible to introduce the convex optimization into humanoid gait generation because the constraint can not be transferred into convex. This paper derives a new ZMP formulation by combini...

This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transforma...

Multi-objective optimization is known as a useful mathematical model in order to investigate some real world problems with connicting objectives, arising from economics, engineering and human decision making. In this paper, a convex composite multi-objective optimization subject to a closed convex set constraint is studied. New rst-order optimality conditions of a weakly eecient solution for th...

CVXPY is a domain-specific language for convex optimization embedded in Python. It allows the user to express convex optimization problems in a natural syntax that follows the math, rather than in the restrictive standard form required by solvers. CVXPY makes it easy to combine convex optimization with high-level features of Python such as parallelism and object-oriented design. CVXPY is availa...

State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothness parameters is necessary. As a special case, we show Nesterov’s accelerate...

I n recent years, we have witnessed technical breakthroughs in a wide variety of topics where the key to success is the use of convex optimization. In fact, convex optimization has now emerged as a major signal processing tool that has made a significant impact on numerous problems previously considered intractable. Considering the foundational nature and potential impact of convex optimization...

This paper is concerned with the development of an integration theory with respect to operator-valued measures which is required in the study o/ certain convex optimization problems. These convex optimization problems in their turn are rigorous ]ormulations o] detection theory in a quantum communication context, which generalise classical (Bayesian) detection theory. The integration theory whic...