Perspective Envelopes for Bilinear Functions
Given the bilinear function f(x,y) = xy, and the constraint x
Constrained Nonlinear Optimization
Optimality and complexity for constrained optimization problems with nonconvex regularization
In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set and the objective function has a nonsmooth, nonconvex regularizer. Such regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding and non-Lipschitz $L_p$ penalties as special cases. Using the theory of the generalized directional … Read more
Copositivity for second-order optimality conditions in general smooth optimization problems
Second-order local optimality conditions involving copositivity of the Hessian of the Lagrangian on the reduced linearization cone have the advantage that there is only a small gap between sufficient (the Hessian is strictly copositive) and necessary (the Hessian is copositive) conditions. In this respect, this is a proper generalization of convexity of the Lagrangian. We … Read more
A cone-continuity constraint qualification and algorithmic consequences
Every local minimizer of a smooth constrained optimization problem satisfies the sequential Approximate Karush-Kuhn-Tucker (AKKT) condition. This optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms. It is natural to ask for conditions on the constraints under which AKKT implies KKT. These conditions will be called Strict Constraint Qualifications … Read more
An external penalty-type method for multicriteria
We propose an extension of the classical real-valued external penalty method to the multicriteria optimization setting. As its single objective counterpart, it also requires an external penalty function for the constraint set, as well as an exogenous divergent sequence of nonnegative real numbers, the so-called penalty parameters, but, differently from the scalar procedure, the vector-valued … Read more
On the Performance of SQP Methods for Nonlinear Optimization
This paper concerns some practical issues associated with the formulation of sequential quadratic programming (SQP) methods for large-scale nonlinear optimization. SQP methods find an approximate solution of a sequence of quadratic programming (QP) subproblems in which a quadratic model of the objective function is minimized subject to the linearized constraints. Extensive numerical results are given … Read more
A trust-funnel method for nonlinear optimization problems with general nonlinear constraints and its application to derivative-free optimization
A trust-funnel method is proposed for solving nonlinear optimization problems with general nonlinear constraints. It extends the one presented by Gould and Toint (Math. Prog., 122(1):155-196, 2010), originally proposed for equality-constrained optimization problems only, to problems with both equality and inequality constraints and where simple bounds are also considered. As the original one, our method … Read more
A Filter SQP Method: Local Convergence and Numerical Results
The work by Gould, Loh, and Robinson [“A filter method with unified step computation for nonlinear optimization”, SIAM J. Optim., 24 (2014), pp. 175–209] established global convergence of a new filter line search method for finding local first-order solutions to nonlinear and nonconvex constrained optimization problems. A key contribution of that work was that the … Read more
Constrained trace-optimization of polynomials in freely noncommuting variables
The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry (RAG). In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation … Read more
Scheduling with Fixed Maintenance, Shared Resources and Nonlinear Feedrate Constraints: a Mine Planning Case Study
Given a short term mining plan, the task for an operational mine planner is to determine how the equipment in the mine should be used each day. That is, how crushers, loaders and trucks should be used to realise the short term plan. It is important to achieve both grade targets (by blending) and maximise … Read more