Notes in TTK4135

To Subscribe, use this Key

Status	Last Update	Fields
Removal Requested	04/22/2024	Equation of a circle in the complex plane
Removal Requested	04/22/2024	Den deriverte til en funksjon
Published	04/23/2024	Convex programming is used to describe a special case of the general constrained optimization problem in whichthe {{c1::objective function}} is convex…
Published	04/23/2024	The function \(f\) is a convex function if its domain \(S\) is a {{c1::convex set}} and if for any two points \(x\) and&…
Published	04/23/2024	Standardized formulation for optimization problems
Published	04/23/2024	When the objective function and all the constraints are linear functions of \(x\), the problem is a {{c1::linear programming problem}}.
Published	04/23/2024	For {{c1::convex programming}} problems (as well as {{c1::linear programming}} problems), {{c2::local}} solutions are also {{c2::global}} solutions.&n…
Published	04/23/2024	A point \(x^\) is a global minimizer if {{c1::\(f(x^) \leq f(x)\) for all \(x\)}}.
Published	04/23/2024	Local minimizer
Published	04/23/2024	If \(f\) is {{c1::twice continuously differentiable}}, we may be able to tell that \(x^*\) is a local minimizer (and possibly a st…
Published	04/23/2024	Positive definite matrix
Published	04/23/2024	Positive semidefinite matrix
Published	04/23/2024	Degenerate basis
Published	04/23/2024	A linear program is degenerate if it has at least one {{c1::degenerate basis}}.
Published	04/23/2024	The two fundamental strategies for moving from the current point \(x_k\) to a new iterate \(x_{k+1}\) in an optimization algorithm
Published	04/23/2024	In the trust region strategy, we find the candidate step \(p\) by approximately solving the following subproblem
Published	04/23/2024	When using trust region: If the candidate solution does not produce a sufficient decrease in the objective function, we conclude that the trust region…
Published	04/23/2024	Line search starts by fixing the {{c1::direction \(p_k\)}} and then identifying an appropriate distance - the {{c1::step length \(\alph…
Published	04/23/2024	Steepest descent direction formula
Published	04/23/2024	Newton direction formula
Published	04/23/2024	The main drawback of the Newton direction is the need for the {{c1::Hessian \(\nabla^2 f(x)\)}}.
Published	04/23/2024	Quasi-Newton search directions differ from Newton's method in that they do not require {{c1::computation of the Hessian}}, and yet they converge at a …
Published	04/23/2024	Definition of an active set
Published	04/23/2024	KKT condition of stationarity
Published	04/23/2024	KKT conditions of primal feasibility
Published	04/23/2024	KKT condition of dual feasibility
Published	04/23/2024	What is the LP trick for handling absolute values in an objective function, and how does it ensure the optimality of a solution?
Published	04/23/2024	What are principal leading minors in a matrix, and how are they determined?
Published	04/23/2024	What is Sylvester's Criterion, and how does it determine the positive definiteness of a symmetric matrix?
Published	04/23/2024	1-norm (sum-norm) for vectors
Published	04/23/2024	2-norm (Euclidean norm) for vectors
Published	04/23/2024	\(\infty\)-norm (max norm) for vectors
Published	04/23/2024	LICQ implies the uniqueness of {{c1::Lagrange multipliers}}.
Published	04/23/2024	The Lagrangian multipliers are the {{c1::"hidden cost"}}/{{c1::"shadow prices"}} of constraints.
Published	04/23/2024	For a well-conditioned matrix, {{c1::small perturbations}} give {{c1::small changes}} in solution.
Published	04/24/2024	Condition number of a matrix (when solving \(Ax=b\))
Published	04/23/2024	A quadratic programming problem is a convex problem if {{c1::\(P \geq 0\)}}.
Published	04/24/2024	Standardized formulation for quadratic programming problems
Published	04/23/2024	There are two sources for no solution for linear programming problems
Published	04/23/2024	The three possible cases for solutions for linear programming problems
Published	04/23/2024	The dual LP problem
Published	04/23/2024	Strong duality
Published	04/23/2024	What is the LQG System matrix?
Published	04/23/2024	What are the Wolfe conditions for step size?
Published	04/23/2024	What is the "easiest" nonlinear programming problem?
Published	04/23/2024	Under which conditions is a QP-problem convex?
Published	04/23/2024	A merit function \(\phi(x; \micro) \)is exact if {{c1::there is a positive scalar \(\micro^*\) such that for any \(\micro > \mi…
Published	04/23/2024	Definition of a feasible set \(\Omega\)
Published	04/23/2024	Formula for the Lagrangian
Published	04/23/2024	When there is an equality constraint, the feasible direction is {{c1::perpendicular to the gradient of the equality constraint}}.
Published	04/24/2024	Given a feasible point \(x\) and the active constraint set \(\mathcal{A}(x)\), the set of linearized feasible directions \(\mathca…
Published	04/26/2024	When the problem is {{c1::convex}}, the KKT conditions are {{c2::necessary}} and {{c2::sufficient}} for a solution.
Published	04/26/2024	Definition of the Hessian
Published	04/26/2024	KKT complementary condition
Published	04/26/2024	A possible solution is a point where there are no directions that are both {{c1::feasible}} and {{c1::descent}} directions.
Published	04/26/2024	What is a tangent cone?
Published	04/26/2024	{{c2::Constraint qualifications}} are needed to rule out special cases where optimal solutions do not {{c1::fulfill the KKT conditions}}.
Published	04/26/2024	For a {{c1::convex}} constrained optimization problem where {{c1::Slater's condition}} is fulfilled, the KKT conditions are {{c2::necessary}} and {{c2…
Published	04/26/2024	{{c2::LICQ}} implies {{c1::Slater's condition}}.
Published	05/01/2024	Second-order sufficient conditionsSuppose that \(x^\) is a local solution and that the LICQ condition is satisfied. Let \(\lambda^\)&…
Published	04/28/2024	A matrix \(A \in \mathbb{R}^{m \times n}\) is a mapping: {{c1:: \(x \in \mathbb{R}^n \) to \(y \in \mathbb{R}^m\)}}
Published	04/28/2024	To use the {{c1::matrix inverse}} for solving linear equation systems is inefficient and numerically unstable.
Published	04/28/2024	Standard form LP
Published	04/28/2024	Lagrangian for a linear programming problem
Published	04/28/2024	The Simplex method generates iterates that are BFP, and converge to a solution if {{c1::there are BFPs}}, and{{c1::one of them is a solution (Bas…
Published	04/28/2024	Fundamental theorem of linear programming
Published	04/28/2024	If an LP is {{c1::bounded}} and {{c1::non-degenerate}}, the Simplex method terminates at a {{c2::BOP}}.
Published	04/28/2024	In {{c1::active set methods}}, one maintains an estimate of the set of {{c2::inequality constraints}} that are {{c2::active at the solution}}.
Published	05/01/2024	Strong duality implies the following relationship between primal and dual
Published	05/02/2024	Will a linear MPC controller always be stable?
Published	05/02/2024	Discrete State Space Equations for linear system with kalman filter feedback
Published	05/02/2024	LTI system with disturbance model
Published	05/02/2024	Equations for linear state estimator with constant disturbance
Published	05/05/2024	Active set method for QPs (simplified)
Published	05/06/2024	If we know which {{c2::inequalities}} are active at the solution, {{c1::QP can be solved as EQP}}.
Published	05/09/2024	Nominal stability
Published	05/09/2024	Robust stability
Published	05/09/2024	Rule of thumb for achieving nominal stability for MPC
Published	05/09/2024	How to ensure that there always is a solution to the MPC open-loop optimization problem?
Status	Last Update	Fields