Solving Optimization Problems using the Matlab Optimization.docx

Solving Optimization Problems using the Matlab Optimization
Toolbox - a Tutorial
TU-Ilmenau, Fakultät für Mathematik und Naturwissenschaften
Dr. Abebe Geletu
December 13, 2007
Contents
1 Introduction to Mathematical Programming 2
1.1 A general Mathematical Programming Problem . . . . . . . . . .
. . . . . . . . . . . . 2
1.1.1 Some Classes of Optimization Problems . . . . . . . . . . . . .
. . . . . . . . . 2
1.1.2 Functions of the Matlab Optimization Toolbox . . . . . . . . .
. . . . . . . . . . 5
2 Linear Programming Problems 6
2.1 Linear programming with MATLAB . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 6
2.2 The Interior Point Method for LP . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 8

2.3 Using linprog to solve LP’s . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 11
2.3.1 Formal problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 11
2.3.2 Approximation of discrete Data by a Curve . . . . . . . . . . .
. . . . . . . . . . 13
3 Quadratic programming Problems 15
3.1 Algorithms Implemented under quadprog.m . . . . . . . . . . . .
. . . . . . . . . . . . 16
3.1.1 Active Set-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 17
3.1.2 The Interior Reflective Method . . . . . . . . . . . . . . . . . . . .
. . . . . . . 19
3.2 Using quadprog to Solve QP Problems . . . . . . . . . . . . . . . .
. . . . . . . . . . . 24
3.2.1 Theoretical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 24
3.2.2 Production model - profit maximization . . . . . . . . . . . . . .
. . . . . . . . 26
4 Unconstrained nonlinear programming 30
4.1 Theory, optimality conditions . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 30
4.1.1 Problems, assumptions, definitions . . . . . . . . . . . . . . . . .
. . . . . . . . 30

4.2 Optimality conditions for smooth unconstrained problems . .
. . . . . . . . . . . . . . 31
4.3 Matlab Function for Unconstrained Optimization . . . . . . . .
. . . . . . . . . . . . . 32
4.4 General descent methods - for differentiable Optimization
Problems . . . . . . . . . . . 32
4.5 The Quasi-Newton Algorithm -idea . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 33
4.5.1 Determination of Search Directions . . . . . . . . . . . . . . . . .
. . . . . . . . 33
4.5.2 Line Search Strategies- determination of the step-length
αk . . . . . . . . . . . 34
4.6 Trust Region Methods - idea . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 35
4.6.1
Solution
of the Trust-Region Sub-problem . . . . . . . . . . . . . . . . . . . . .
36
4.6.2 The Trust Sub-Problem under Considered in the Matlab
Optimization toolbox . 38

4.6.3 Calling and Using fminunc.m to Solve Unconstrained
Problems . . . . . . . . . 39
4.7 Derivative free Optimization - direct (simplex) search
methods . . . . . . . . . . . . . . 45
1
Chapter 1
Introduction to Mathematical
Programming
1.1 A general Mathematical Programming Problem
f(x) −→ min (max)
subject to
x ∈ M.
(O)
The function f : Rn → R is called the objective function and the

set M ⊂ Rn is the feasible set of (O).
Based on the description of the function f and the feasible set
M, the problem (O) can be classified
as linear, quadratic, non-linear, semi-infinite, semi-definite,
multiple-objective, discrete optimization
problem etc1.
1.1.1 Some Classes of Optimization Problems
Linear Programming
If the objective function f and the defining functions of M are
linear, then (O) will be a linear
optimization problem.
General form of a linear programming problem:
c⊤ x −→ min (max)
s.t.
Ax = a
Bx ≤ b
lb ≤ x ≤ ub;
(LO)

i.e. f(x) = c⊤ x and M = {x ∈ Rn | Ax = a,Bx ≤ b,lb ≤ x ≤ ub}.
Under linear programming problems are such practical problems
like: linear discrete Chebychev ap-
proximation problems, transportation problems, network flow
problems,etc.
1The terminology mathematical programming is being currently
contested and many demand that problems of the form
(O) be always called mathematical optimization problems. Here,
we use both terms alternatively.
2
Sou Ying
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Sou Ying
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3
Quadratic Programming
1
2
xT Qx + q⊤ x → min
s.t.
Ax = a
Bx ≤ b
x ≥ u
x ≤ v
(QP)
Here the objective function f(x) = 12x⊤ Qx + q⊤ x is a quadratic
function, while the feasible set
M = {x ∈ Rn | Ax = a,Bx ≤ b,u ≤ x ≤ v} is defined using linear
functions.
One of the well known practical models of quadratic
optimization problems is the least squares ap-

proximation problem; which has applications in almost all fields
of science.
Non-linear Programming Problem
The general form of a non-linear optimization problem is
f (x) −→ min (max)
subject to
equality constraints: gi (x) = 0, i ∈ {1, 2, . . . ,m}
inequality constraints: gj (x) ≤ 0, j ∈ {m + 1,m + 2, . . . ,m + p}
box constraints: uk ≤ xk ≤ vk, k = 1, 2, . . . ,n;
(NLP)
where, we assume that all the function are smooth, i.e. the
functions
f, gl : U −→ R l = 1, 2, . . . ,m + p
are sufficiently many times differentiable on the open subset U
of Rn. The feasible set of (NLP) is
given by

M = {x ∈ Rn | gi(x) = 0, i = 1, 2, . . . ,m; gj (x) ≤ 0,j = m + 1,m
+ 2, . . . ,m + p} .
We also write the (NLP) in vectorial notation as
f (x) → min (max)
h (x) = 0
g (x) ≤ 0
u ≤ x ≤ v.
Problems of the form (NLP) arise frequently in the numerical
solution of control problems, non-linear
approximation, engineering design, finance and economics,
signal processing, etc.
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4

Semi-infinite Programming
f(x) → min
s.t.
G(x,y) ≤ 0,∀ y ∈ Y ;
hi(x) = 0, i = 1, . . . ,p;
gj (x) ≤ 0,j = 1, . . . ,q;
x ∈ Rn;
Y ⊂ Rm.
(SIP)
Here, f,hi,gj : R
n → R, i ∈ {1, . . . ,p},j ∈ {1, . . . ,q} are smooth functions; G :
Rn × Rm → R is such
that, for each fixed y ∈ Y , G(·,y) : Rn → R is smooth and, for
each fixed x ∈ Rn,G(x, ·) : Rm → R is
smooth; furthermore, Y is a compact subset of Rm. Sometimes,
the set Y can also be given as
Y = {y ∈ Rm | uk(y) = 0,k = 1, . . . ,s1; vl(y) ≤ 0, l = 1, . . . ,s2}

with smooth functions uk,vl : R
m → R,k ∈ {1, . . . ,s1}, l ∈ {1, . . . ,s2}.
The problem (SIP) is called semi-infinite, since its an
optimization problem with finite number of vari-
ables (i.e. x ∈ Rn) and infinite number of constraints (i.e.
G(x,y) ≤ 0,∀ y ∈ Y ).
One of the well known practical models of (SIP) is the
continuous Chebychev approximation problem.
This approximation problem can be used for the approximation
of functions by polynomials, in filter
design for digital signal processing, spline approximation of
robot trajectory
Multiple-Objective Optimization
A multiple objective Optimization problem has a general form
min(f1(x),f1(x), . . . ,fm(x))
s.t.

x ∈ M;
(MO)
where the functions fk : R
n → R,k = 1, . . . ,m are smooth and the feasible set M is
defined in terms of
linear or non-linear functions. Sometimes, this problem is also
alternatively called multiple-criteria,
vector optimization, goal attainment or multi-decision analysis
problem. It is an optimization
problem with more than one objective function (each such
objective is a criteria). In this sense,
(LO),(QP)(NLO) and (SIP) are single objective (criteria)
optimization problems. If there are only two
objective functions in (MO), then (MO) is commonly called to
be a bi-criteria optimization problem.
Furthermore, if each of the functions f1, . . . ,fm are linear and
M is defined using linear functions,

then (MO) will be a linear multiple-criteria optimization
problem; otherwise, it is non-linear.
For instance, in a financial application we may need, to
maximize revenue and minimize risk at the
same time, constrained upon the amount of our investment.
Several engineering design problems can
also be modeled into (MO). Practical problems like autonomous
vehicle control, optimal truss design,
antenna array design, etc are very few examples of (MO).
In real life we may have several objectives to arrive at. But,
unfortunately, we cannot satisfy all
our objectives optimally at the same time. So we have to find a
compromising solution among all
our objectives. Such is the nature of multiple objective
optimization. Thus, the minimization (or

5
maximization) of several objective functions can not be done in
the usual sense. Hence, one speaks
of so-called efficient points as solutions of the problem. Using
special constructions involving the
objectives, the problem (MO) can be reduced to a problem with
a single objective function.
1.1.2 Functions of the Matlab Optimization Toolbox
Linear and Quadratic Minimization problems.
linprog - Linear programming.
quadprog - Quadratic programming.
Nonlinear zero finding (equation solving).
fzero - Scalar nonlinear zero finding.
fsolve - Nonlinear system of equations solve (function solve).

Linear least squares (of matrix problems).
lsqlin - Linear least squares with linear constraints.
lsqnonneg - Linear least squares with nonnegativity constraints.
Nonlinear minimization of functions.
fminbnd - Scalar bounded nonlinear function minimization.
fmincon - Multidimensional constrained nonlinear
minimization.
fminsearch - Multidimensional unconstrained nonlinear
minimization, by Nelder-Mead direct
search method.
fminunc - Multidimensional unconstrained nonlinear
minimization.
fseminf - Multidimensional constrained minimization, semi-
infinite constraints.
Nonlinear least squares (of functions).

lsqcurvefit - Nonlinear curvefitting via least squares (with
bounds).
lsqnonlin - Nonlinear least squares with upper and lower
bounds.
Nonlinear minimization of multi-objective functions.
fgoalattain - Multidimensional goal attainment optimization
fminimax - Multidimensional minimax optimization.
Sou Ying
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Chapter 2
Linear Programming Problems
2.1 Linear programming with MATLAB
For the linear programming problem

c⊤ x −→ min
s.t.
Ax ≤ a
Bx = b
lb ≤ x ≤ ub;
(LP)
MATLAB: The program linprog.m is used for the minimization
of problems of the form (LP).
Once you have defined the matrices A, B, and the vectors
c,a,b,lb and ub, then you can call linprog.m
to solve the problem. The general form of calling linprog.m is:
[x,fval,exitflag,output,lambda]=linprog(f,A,a,B,b,lb,ub,x0,optio
ns)
Input arguments:
c coefficient vector of the objective
A Matrix of inequality constraints

a right hand side of the inequality constraints
B Matrix of equality constraints
b right hand side of the equality constraints
lb,[ ] lb ≤ x : lower bounds for x, no lower bounds
ub,[ ] x ≤ ub : upper bounds for x, no upper bounds
x0 Startvector for the algorithm, if known, else [ ]
options options are set using the optimset funciton, they
determine what algorism to use,etc.
Output arguments:
x optimal solution
fval optimal value of the objective function
exitflag tells whether the algorithm converged or not, exitflag >
0 means convergence
output a struct for number of iterations, algorithm used and
PCG iterations(when LargeScale=on)

lambda a struct containing lagrange multipliers corresponding
to the constraints.
6
7
Setting Options
The input argument options is a structure, which contains
several parameters that you can use with
a given Matlab optimization routine.
For instance, to see the type of parameters you can use with the
linprog.m routine use
>>optimset(’linprog’)
Then Matlab displays the fileds of the structure options.
Accordingly, before calling linprog.m you

can set your preferred parameters in the options for linprog.m
using the optimset command as:
>>options=optimset(’ParameterName1’,value1,’ParameterName
2’,value2,...)
where ’ParameterName1’,’ParameterName2’,... are those you
get displayed when you use
optimset(’linprog’). And value1, value2,... are their
corresponding values.
The following are parameters and their corresponding values
which are frequently used with linprog.m:
Parameter Possible Values
’LargeScale’ ’on’,’off’
’Simplex’ ’on’,’off’
’Display’ ’iter’,’final’,’off’
’Maxiter’ Maximum number of iteration

’TolFun’ Termination tolerance for the objective function
’TolX’ Termination tolerance for the iterates
’Diagnostics’ ’on’ or ’off’ (when ’on’ prints diagnostic
information about the objective function)
Algorithms under linprog
There are three type of algorithms that are being implemented
in the linprog.m:
• a simplex algorithm;
• an active-set algorithm;
• a primal-dual interior point method.
The simplex and active-set algorithms are usually used to solve
medium-scale linear programming
problems. If any one of these algorithms fail to solve a linear
programming problem, then the problem
at hand is a large scale problem. Moreover, a linear

programming problem with several thousands of
variables along with sparse matrices is considered to be a large-
scale problem. However, if coefficient
matrices of your problem have a dense matrix structure, then
linprog.m assumes that your problem
is of medium-scale.
By default, the parameter ’LargeScale’ is always ’on’. When
’LargeScale’ is ’on’, then linprog.m
uses the primal-dual interior point algorithm. However, if you
want to set if off so that you can solve
a medium scale problem , then use
>>options=optimset(’LargeScale’,’off’)
8
In this case linprog.m uses either the simplex algorithm or the

active-set algorithm. (Nevertheless,
recall that the simplex algorithm is itself an active-set strategy).
If you are specifically interested to use the active set algorithm,
then you need to set both the param-
eters ’LargeScale’ and ’Simplex’, respectively, to ’off’:
>>options=optimset(’LargeScale’,’off’,’Simplex’,’off’)
Note: Sometimes, even if we specified ’LargeScale’ to be ’off’,
when a linear programming problem
cannot be solved with a medium scale algorithm, then linprog.m
automatically switches to the large
scale algorithm (interior point method).
2.2 The Interior Point Method for LP
Assuming that the simplex method already known, we find this
section a brief discussion on the
primal-dual interior point method for (LP).

Let A ∈ Rm×n,a ∈ Rm,B ∈ Rp×n,b ∈ Rp. Then, for the linear
programming problem
c⊤ x −→ min
s.t.
Ax ≤ a
Bx = b
lb ≤ x ≤ ub;
(LP)
if we set x̃ = x− lb we get
c⊤x̃− c⊤ lb −→ min
s.t.
Ax̃ ≤ a−A(lb)
Bx̃ = b−B(lb)
0 ≤ x̃ ≤ ub− lb;
(LP)
Now, by adding slack variables y ∈ Rm and s ∈ Rn (see below),
we can write (LP) as

c⊤x̃−c⊤ lb −→ min
s.t.
Ax̃ +y = a−A(lb)
Bx̃ = b−B(lb)
x̃ +s = ub− lb
x̃ ≥ 0, y ≥ 0, s ≥ 0.
(LP)
Thus, using a single matrix for the constraints, we have
c⊤x̃− c⊤ lb −→ min
A Im Om×n
B Op×m Op×n
In On×n In

x̃
y
s
a−A(lb)
b−B(lb)
ub− lb
x̃ ≥ 0,y ≥ 0,s ≥ 0.
9
Since, a constant in the objective does not create a difficulty,
we assume w.l.o.g that we have a problem

of the form
c⊤ x −→ min
s.t.
Ax = a
x ≥ 0.
(LP’)
In fact, when you call linprog.m with the original problem (LP),
this transformation will be done by
Matlab internally. The aim here is to briefly explain the
algorithm used, when you set the LargeScale
parameter to ’on’ in the options of linprog.
Now the dual of (LP’) is the problem
b⊤ w −→ max
s.t.
A⊤ w ≤ c
w ∈ Rm.
(LPD)

Using a slack variable s ∈ Rn we have
b⊤ w −→ max
s.t.
A⊤ w + s = c
w ∈ Rm, s ≥ 0 .
(LPD)
The problem (LP’) and (LPD) are called primal-dual pairs.
Optimality Condition
It is well known that a vector (x∗ ,w∗ ,s∗ ) is a solution of the
primal-dual if and only if it satisfies the
Karush-Kuhn-Tucker (KKT) optimlaity condition. The KKT
conditions here can be written as
A⊤ w + s = c
Ax = a
xisi = 0, i = 1, . . . ,n(Complemetarity conditions)

(x,y) ≥ 0.
This system can be written as
F(x,w,s) =
A⊤ w + s− c
Ax−a
XSe
(x,s) ≥ 0, (2.2)
where X = diag(x1,x2, . . . ,xn),S = diag(s1,s2, . . . ,sn) ∈ R
n×n,e = (1, 1, . . . , 1)⊤ ∈ Rn.
Primal-dual interior point methods generate iterates (xk,wk,sk)
that satisfy the system (2.1) & (2.2)
so that (2.2) is satisfied strictly; i.e. xk > 0,sk > 0. That is, for
each k, (xk,sk) lies in the interior
of the nonnegative-orthant. Thus the naming of the method as

interior point method. Interior point
methods use a variant of the Newton method for the system
(2.1) & (2.2).
10
Central Path
Let τ > 0 be a parameter. The central path is a curve C which is
the set of all points (x(τ),w(τ),s(τ)) ∈
C that satisfy the parametric system :
A⊤ w + s = c,
Ax = b,
xsi = τ,i = 1, . . . ,n
(x,s) > 0.
This implies C is the set of all points (x(τ),w(τ),s(τ)) that
satisfy

F(x(τ),w(τ),s(τ)) =
0
0
τe
Obviously, if we let τ ↓ 0, the the system (2.3) goes close to the
system (2.1) & (2.2).
Hence, theoretically, primal-dual algorithms solve the system
J(x(τ),w(τ),s(τ))
△x(τ)
△w(τ)
△s(τ)

0
0
−XSe + τe
to determine a search direction (△x(τ),△w(τ),△s(τ)), where
J(x(τ),w(τ),s(τ)) is the Jacobian of
F(x(τ),w(τ),s(τ)). And the new iterate will be
(x+(τ),w+(τ),s+(τ)) = (x(τ),w(τ),s(τ)) + α(△x(τ),△w(τ),△s(τ)),
where α is a step length, usually α ∈ (0, 1], chosen in such a
way that (x+(τ),w+(τ),s+(τ) ∈ C.
However, practical primal-dual interior point methods use τ =

σµ, where σ ∈ [0, 1] is a constant and
µ =
x⊤ s
n
The term x⊤ s is the duality gap between the primal and dual
problems. Thus, µ is the measure of
the (average) duality gap. Note that, in general, µ ≥ 0 and µ = 0
when x and s are primal and dual
optimal, respectively.
Thus the Newton step (△x(µ),△w(µ),△s(µ)) is determined by
solving:
On A
⊤ In
A On×m Om×n
S On×m X

△x(τ)
△w(τ)
△s(τ)
0
0
−XSe + σµe
The Newton step (△x(µ),△w(µ),△s(µ)) is also called centering
direction that pushes the iterates
(x+(µ),w+(µ),s+(µ) towards the central path C along which the

algorithm converges more rapidly.
The parameter σ is called the centering parameter. If σ = 0, then
the search direction is known to
be an affine scaling direction.
Primal-Dual Interior Point Algorithm
11
Step 0: Start with (x0,w0,s0) with (x0,s0) > 0, k = 0
Step k: choose σk ∈ [0, 1], set µk = (x
k)⊤ sk/n and solve
On A
⊤ In
A On×m Om×n
S On×m X

△xk
△wk
△sk
0
0
−XkSke + σkµke
set

(xk+1,wk+1,sk+1) ← (xk,wk,sk) + αk(△x
k,△wk,△sk)
choosing αk ∈ [0, 1] so that (x
k+1,sk+1) > 0.
If (convergence) STOP else set k ← k + 1 GO To Step k.
The Matlab LargeSclae option of linprog.m uses a predicator-
corrector like method of Mehrotra to
guarantee that (xk,sk) > 0 for each k,k = 1, 2, . . .
Predicator Step: A search direction (a predicator step) dkp =
(△x
k,△wk,△sk) is obtained by solving
the non-parameterized system (2.1) & (2.2).
Corrector Step: For a centering parameter σ is obtained from
dkc = [F
⊤(xk,wk,sk)]−1F(xk + ⊤xk,wk⊤wk,sk + ⊤sk) −σê

where ê ∈ Rn+m+n, whose last n components are equal to 1.
Iteration: for a step length α ∈ (0, 1]
(xk+1,wk+1,sk+1) = (xk,wk,sk) + α(dkp + d
k
c ).
2.3 Using linprog to solve LP’s
2.3.1 Formal problems
1. Solve the following linear optimization problem using
linprog.m.
2x1+ 3x2 → max
s.t.
x1+ 2x2 ≤ 8
2x1+ x2 ≤ 10
x2 ≤ 3
x1, x2 ≥ 0
For this problem there are no equality constraints and box

constraints, i.e. B=[],b=[],lb=[] and
ub=[]. Moreover,
>>c=[-2,-3]’; % linprog solves minimization problems
>>A=[1,2;2,1;0,1];
>>a=[8,10,3]’;
>>options=optimset(’LargeScale’,’off’);
12
i) If you are interested only on the solution, then use
>>xsol=linprog(c,A,b,[],[],[],[],[],options)
ii) To see if the algorithm really converged or not you need to
access the exit flag through:
>>[xsol,fval,exitflag]=linprog(c,A,a,[],[],[],[],[],options)

iii) If you need the Lagrange multipliers you can access them
using:
>>[xsol,fval,flag,output,LagMult]=linprog(c,A,a,[],[],[],[],[],opt
ions)
iv) To display the iterates you can use:
>>xsol=linprog(c,A,a,[],[],[],[],[],optimset(’Display’,’iter’))
2. Solve the following LP using linprog.m
c⊤ x −→ max
Ax = a
Bx ≥ b
Dx ≤ d
lb ≤ x ≤ lu
where
(A|a) =
(
1 1 1 1 1 1

5 0 −3 0 1 0
∣ ∣ ∣ ∣
10
15
)
, (B|b) =
1 2 3 0 0 0
0 1 2 3 0 0
0 0 1 2 3 0
0 0 0 1 2 3
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
5
7
8
8

(D|d) =
(
3 0 0 0 −2 1
0 4 0 −2 0 3
∣ ∣ ∣ ∣
5
7
)
, lb =
−2
0
−1
−1
−5
1

, lu =
7
2
2
3
4
10
,c =
1
−2

3
−4
5
−6
.
When there are large matrices, it is convenient to write m files.
Thus one possible solution will
be the following:
function LpExa2
A=[1,1,1,1,1,1;5,0,-3,0,1,0]; a=[10,15]’;
B1=[1,2,3,0,0,0; 0,1,2,3,0,0;... 0,0,1,2,3,0;0,0,0,1,2,3];
b1=[5,7,8,8];b1=b1(:);
D=[3,0,0,0,-2,1;0,4,0,-2,0,3]; d=[5,7]; d=d(:);

lb=[-2,0,-1,-1,-5,1]’; ub=[7,2,2,3,4,10]’;
c=[1,-2,3,-4,5,-6];c=c(:);
B=[-B1;D]; b=[-b1;d];
[xsol,fval,exitflag,output]=linprog(c,A,a,B,b,lb,ub)
fprintf(’%s %s n’, ’Algorithm Used: ’,output.algorithm);
disp(’=================================’);
disp(’Press Enter to continue’); pause
options=optimset(’linprog’);
13
options =
optimset(options,’LargeScale’,’off’,’Simplex’,’on’,’Display’,’it
er’);
[xsol,fval,exitflag]=linprog(c,A,a,B,b,lb,ub,[],options)

fprintf(’%s %s n’, ’Algorithm Used: ’,output.algorithm);
fprintf(’%s’,’Reason for termination:’)
if (exitflag)
fprintf(’%s n’,’ Convergence.’);
else
fprintf(’%s n’,’ No convergence.’);
end
Observe that for the problem above the simplex algorithm does
not work properly. Hence,
linprog.m uses automatically the interior point method.
2.3.2 Approximation of discrete Data by a Curve
Solve the following discrete approximation problem and plot the
approximating curve.

Suppose the measurement of a real process over a 24 hours
period be given by the following table
with 14 data values:
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14
ti 0 3 7 8 9 10 12 14 16 18 19 20 21 23
ui 3 5 5 4 3 6 7 6 6 11 11 10 8 6
The values ti represent time and ui’s are measurements.
Assuming there is a mathematical connection
between the variables t and u, we would like to determine the
coefficients a,b,c,d,e ∈ R of the
function
u(t) = at4 + bt3 + ct2 + dt + e
so that the value of the function u(ti) could best approximate
the discrete value ui at ti, i = 1, . . . , 14
in the Chebychev sense. Hence, we need to solve the Chebyshev
approximation problem
(CA)

maxi=1,...,14
∣ ∣ ui −
(
at4i + bt
3
i + ct
2
i + dti + e
)∣ ∣ → min
s.t. a,b,c,d,e ∈ R

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