Topic
Convex optimization
About: Convex optimization is a research topic. Over the lifetime, 24906 publications have been published within this topic receiving 908795 citations. The topic is also known as: convex optimisation.
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01 Jun 1985
TL;DR: Methods with subgradient locality measures for minimizing nonconvex functions and methods of feasible directions for non Convex constrained problems for convex constrained minimization problems are described.
Abstract: Fundamentals.- Aggregate subgradient methods for unconstrained convex minimization.- Methods with subgradient locality measures for minimizing nonconvex functions.- Methods with subgradient deletion rules for unconstrained nonconvex minimization.- Feasible point methods for convex constrained minimization problems.- Methods of feasible directions for nonconvex constrained problems.- Bundle methods.- Numerical examples.
503 citations
TL;DR: This work presents a subgradient algorithm for generating approximate saddle points and provides per-iteration convergence rate estimates on the constructed solutions, and focuses on Lagrangian duality, where it is shown this algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily.
Abstract: We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.
497 citations
TL;DR: The overall system is proved to fulfill the constraints, be asymptotically stable, and exhibit an offset-free tracking behavior, provided that an admissibility condition on the initial state is satisfied.
Abstract: A method based on conceptual tools of predictive control is described for solving set-point tracking problems wherein pointwise-in-time input and/or state inequality constraints are present. It consists of adding to a primal compensated system a nonlinear device, called command governor (CG), whose action is based on the current state, set-point, and prescribed constraints. The CG selects at any time a virtual sequence among a family of linearly parameterized command sequences, by solving a convex constrained quadratic optimization problem, and feeds the primal system according to a receding horizon control philosophy. The overall system is proved to fulfill the constraints, be asymptotically stable, and exhibit an offset-free tracking behavior, provided that an admissibility condition on the initial state is satisfied. Though the CG can be tailored for the application at hand by appropriately choosing the available design knobs, the required online computational load for the usual case of affine constraints is well tempered by the related relatively simple convex quadratic programming problem.
496 citations
29 Jun 1994
TL;DR: In this article, the authors give a historical view of linear matrix inequalities in control and system theory, and it appears that linear matrices have been involved in some of the major events of control theory.
Abstract: The purpose of this paper is to give a historical view of linear matrix inequalities in control and system theory. Not surprisingly, it appears that LMIs have been,involved in some of the major events of control theory. With the advent of powerful convex optimization techniques, LMIs are now about to become an important practical tool for future control applications.
484 citations
TL;DR: This work presents a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing as a finite-dimensional convex optimization problem as a second-order cone programming problem.
Abstract: We present a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing. Our main contribution is the formulation of the trajectory optimization problem, which has nonconvex control constraints, as a finite-dimensional convex optimization problem, specifically as a second-order cone programming problem. Second-order cone programming is a subclass of convex programming, and there are efficient second-order cone programming solvers with deterministic convergence properties. Consequently, the resulting guidance algorithm can potentially be implemented onboard a spacecraft for real-time applications.
482 citations