Constrained Nonlinear Optimization

Active-set identification with complexity guarantees of an almost cyclic 2-coordinate descent method with Armijo line search

Categories Constrained Nonlinear Optimization

In this paper, it is established finite active-set identification of an almost cyclic 2-coordinate descent method for problems with one linear coupling constraint and simple bounds. First, general active-set identification results are stated for non-convex objective functions. Then, under convexity and a quadratic growth condition (satisfied by any strongly convex function), complexity results on the … Read more

A Penalty-free Infeasible Approach for a Class of Nonsmooth Optimization Problems over the Stiefel Manifold

Categories Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , , ,

Transforming into an exact penalty function model with convex compact constraints yields efficient infeasible approaches for optimization problems with orthogonality constraints. For smooth and L21-norm regularized cases, these infeasible approaches adopt simple and orthonormalization-free updating schemes and show high efficiency in some numerical experiments. However, to avoid orthonormalization while enforcing the feasibility of the final … Read more

Controllable Transmission Networks UnderDemand Uncertainty with Modular FACTS

Categories Constrained Nonlinear Optimization, Optimization Software and Modeling Systems, Scheduling Tags , ,

The transmission system operators (TSOs) are responsible to provide secure and efficient access to the transmission system for all stakeholders. This task is gradually getting challenging due to the demand growth, demand uncertainty, rapid changes in generation mix, and market policies. Traditionally, the TSOs try to maximize the technical performance of the transmission network via … Read more

A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization

Categories Constrained Nonlinear Optimization Tags , , ,

In this paper, we propose an algorithmic framework dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous … Read more

A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

An iterative procedure to find the optimal solutions of general fractional nonlinear delay systems with quadraticperformance indices is introduced. The derivatives of state equations are understood in the Caputo sense. By presenting and applying a general framework, we use the Chebyshev wavelet method developed for fractional linear optimal control to convert fractional nonlinear optimal control … Read more

Beyond local optimality conditions: the case of maximizing a convex function

Categories Constrained Nonlinear Optimization, Global Optimization Tags , ,

In this paper, we design an algorithm for maximizing a convex function over a convex feasible set. The algorithm consists of two phases: in phase 1 a feasible solution is obtained that is used as an initial starting point in phase 2. In the latter, a biconvex problem equivalent to the original problem is solved … Read more

How do exponential size solutions arise in semidefinite programming?

Categories Constrained Nonlinear Optimization, Convex Optimization, Semi-definite Programming

Semidefinite programs (SDPs) are some of the most popular and broadly applicable optimization problems to emerge in the last thirty years. A curious pathology of SDPs, illustrated by a classical example of Khachiyan, is that their solutions may need exponential space to even write down. Exponential size solutions are the main obstacle to solve a … Read more

A Matrix-Free Trust-Region Newton Algorithm for Convex-Constrained Optimization

Categories Constrained Nonlinear Optimization, Infinite Dimensional Optimization, Nonlinear Optimization Tags , , , ,

We describe a matrix-free trust-region algorithm for solving convex-constrained optimization problems that uses the spectral projected gradient method to compute trial steps. To project onto the intersection of the feasible set and the trust region, we reformulate and solve the dual projection problem as a one-dimensional root finding problem. We demonstrate our algorithm’s performance on … Read more

ALESQP: An augmented Lagrangian equality-constrained SQP method for optimization with general constraints

Categories Constrained Nonlinear Optimization, Infinite Dimensional Optimization, Nonlinear Optimization

We present a new algorithm for infinite-dimensional optimization with general constraints, called ALESQP. In short, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equality-constrained nonlinear optimization subproblems at every iteration. The subproblems are solved using a matrix-free trust-region sequential quadratic programming (SQP) method that takes advantage of iterative, i.e., inexact linear … Read more

Weak notions of nondegeneracy in nonlinear semidefinite programming

Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , ,

The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, $\ell$-dimensional kernel of the constraint matrix, by the linear independence of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this … Read more