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Last Update
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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&…
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04/23/2024
Standardized formulation for optimization problems
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04/23/2024
When the objective function and all the constraints are linear functions of \(x\), the problem is a {{c1::linear programming problem}}.
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04/23/2024
For {{c1::convex programming}} problems (as well as {{c1::linear programming}} problems), {{c2::local}} solutions are also {{c2::global}} solutions.&n…
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04/23/2024
A point \(x^*\) is a global minimizer if {{c1::\(f(x^*) \leq f(x)\) for all \(x\)}}.
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04/23/2024
Local minimizer
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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…
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04/23/2024
Positive definite matrix
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04/23/2024
Positive semidefinite matrix
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04/23/2024
Degenerate basis
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04/23/2024
A linear program is degenerate if it has at least one {{c1::degenerate basis}}.
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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
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04/23/2024
In the trust region strategy, we find the candidate step \(p\) by approximately solving the following subproblem
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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…
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04/23/2024
Line search starts by fixing the {{c1::direction \(p_k\)}} and then identifying an appropriate distance - the {{c1::step length \(\alph…
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04/23/2024
Steepest descent direction formula
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04/23/2024
Newton direction formula
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04/23/2024
The main drawback of the Newton direction is the need for the {{c1::Hessian \(\nabla^2 f(x)\)}}.
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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 …
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04/23/2024
Definition of an active set
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04/23/2024
KKT condition of stationarity
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04/23/2024
KKT conditions of primal feasibility
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04/23/2024
KKT condition of dual feasibility
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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?
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04/23/2024
What is Sylvester's Criterion, and how does it determine the positive definiteness of a symmetric matrix?
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04/23/2024
1-norm (sum-norm) for vectors
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04/23/2024
2-norm (Euclidean norm) for vectors
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04/23/2024
\(\infty\)-norm (max norm) for vectors
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04/23/2024
LICQ implies the uniqueness of {{c1::Lagrange multipliers}}.
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04/23/2024
The Lagrangian multipliers are the {{c1::"hidden cost"}}/{{c1::"shadow prices"}} of constraints.
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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 \(Ax=b\))
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04/23/2024
A quadratic programming problem is a convex problem if {{c1::\(P \geq 0\)}}.
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04/24/2024
Standardized formulation for quadratic programming problems
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04/23/2024
There are two sources for no solution for linear programming problems
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04/23/2024
The three possible cases for solutions for linear programming problems
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04/23/2024
The dual LP problem
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04/23/2024
Strong duality
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04/23/2024
What is the LQG System matrix?
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04/23/2024
What are the Wolfe conditions for step size?
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04/23/2024
What is the "easiest" nonlinear programming problem?
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04/23/2024
Under which conditions is a QP-problem convex?
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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…
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04/23/2024
Definition of a feasible set \(\Omega\)
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04/23/2024
Formula for the Lagrangian
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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.
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04/26/2024
Definition of the Hessian
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04/26/2024
KKT complementary condition
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04/26/2024
A possible solution is a point where there are no directions that are both {{c1::feasible}} and {{c1::descent}} directions.
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04/26/2024
What is a tangent cone?
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04/26/2024
{{c2::Constraint qualifications}} are needed to rule out special cases where optimal solutions do not {{c1::fulfill the KKT conditions}}.
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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
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04/28/2024
Lagrangian for a linear programming problem
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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
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04/28/2024
If an LP is {{c1::bounded}} and {{c1::non-degenerate}}, the Simplex method terminates at a {{c2::BOP}}.
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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
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05/02/2024
LTI system with disturbance model
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05/02/2024
Equations for linear state estimator with constant disturbance
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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
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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?
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Last Update
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