Gabriele Farina - Central path and interior-point methods

MIT 6.7220/15.084 — Nonlinear Optimization (Spring ‘25) May 6-8th 2025
Lecture 20
Central path and interior-point methods
Instructor: Prof. Gabriele Farina ( gfarina@mit.edu)★
Having laid the foundations of self-concordant functions, we are ready to see one of the
most important applications of these functions: interior-point methods.
Since we will be working extensively with self-concordant functions, we will make the blanket
assumption that Ω is an open, convex, nonempty set.
L20.1 Path-following interior-point methods: chasing the cen-
tral path
Consider a problem of the form
min
𝑥
s.t.
⟨𝑐, 𝑥⟩
𝑥 ∈ Ω,
where 𝑐 ∈ ℝ𝑛 and Ω denotes the closure of the
open, convex, and nonempty set Ω ⊆ ℝ𝑛.
Unlike iterative methods that project onto the
feasible set (such as for example the projected gra-
dient descent and the mirror descent algorithm),
interior-point methods work by constructing a
sequence of feasible points in Ω, whose limit is
the solution to the problem. To do so, interior-
point methods consider a sequence of optimization
problems with objective
𝛾⟨𝑐, 𝑥⟩ + 𝑓(𝑥),
where 𝛾 ≥ 0 is a parameter and 𝑓 is a strongly
nondegenerate self-concordant function on Ω.
Ω
Central path
𝜋(𝛾)
𝛾 = 0
𝛾 = +∞
Figure: The central path traced by
the sequence of solutions to the regu-
larized problem arg min{−𝛾 ⋅ (𝑥 + 𝑦) +
𝑓(𝑥) : 𝑥 ∈ Ω}, for increasing values of 𝛾 ≥
0. The self-concordant function 𝑓 is the
polyhedral barrier. The red dot, corre-
sponding to the solution at 𝛾 = 0, is
called analytic center.
As we saw in Lecture 19, self-concordant functions shoot to infinity at the boundary of their
domain, and hence the minimizer of the self-concordant function will guarantee that the
solution is in the interior of the feasible set. The parameter 𝛾 is increased over time: as 𝛾
grows, the original objective function ⟨𝑐, 𝑥⟩ becomes the dominant term, and the solution to
the regularized problem will approach more and more the boundary. The path of solutions
traced by the regularized problems is called the central path.
Definition L20.1 (Central path). Let 𝑓 : Ω → ℝ be a lower-bounded strongly nondegen-
erate self-concordant function. The central path is the curve 𝜋 parameterized over 𝛾 ≥
0, traced by the solutions¹ to the regularized optimization problem
𝜋(𝛾) ≔ arg min
𝑥
𝛾⟨𝑐, 𝑥⟩ + 𝑓(𝑥)
s.t. 𝑥 ∈ Ω.
L20.1.1 Barriers and their complexity parameter
As it turns out, the performance of path-following interior-point methods depends crucially
on a parameter of the strongly nondegenerate self-concordant function used, which is called
the complexity parameter of the function.
Definition L20.2 (Complexity parameter). The complexity parameter of a strongly
nondegenerate self-concordant function 𝑓 : Ω → ℝ is defined as the supremum of the
intrinsic squared norm of the second-order descent direction (Newton step) at any point
in the domain, that is,
𝜃𝑓 ≔ sup
𝑥∈Ω
‖𝑛(𝑥)‖2
𝑥.
Theorem L20.1 ([NN94], Corollary 2.3.3). The complexity parameter of a strongly
nondegenerate self-concordant function is at least 1.
We reserve the term barrier for only those self-concordant functions for which the complexity
parameter is finite, as we make formal next.
Definition L20.3 (Barrier function). A strongly nondegenerate self-concordant barrier
(for us, simply barrier) is a strongly nondegenerate self-concordant function 𝑓 whose
complexity parameter is finite.
For example, in the case of the log barrier for the positive orthant, we can bound the
complexity parameter as follows.
Example L20.1. The logarithmic barrier for the positive orthant ℝ𝑛
>0, defined as
𝑓 : ℝ𝑛
>0 → ℝ where 𝑓(𝑥) = − ∑
𝑛
𝑖=1
log(𝑥𝑖)
has complexity parameter 𝜃𝑓 = 𝑛.
Solution. The Hessian of the logarithmic barrier is
∇2𝑓(𝑥) = diag( 1
𝑥2
1
, ..., 1
𝑥2
𝑛
),
and the Newton step is
𝑛(𝑥) = −[∇2𝑓(𝑥)]−1∇𝑓(𝑥) =
(
CCD𝑥1
⋮
𝑥𝑛)
GGH.
Hence, the intrinsic norm of the Newton step satisfies
‖𝑛(𝑥)‖2
𝑥 = 𝑛(𝑥)⊤[∇2𝑓(𝑥)]𝑛(𝑥) = ∑
𝑛
𝑖=1
1
𝑥2
𝑖
𝑥2
𝑖 = 𝑛
as we wanted to show. □
However, not all self-concordant functions are barriers.
Example L20.2. The function 𝑓(𝑥) = 𝑥 − log(𝑥) is strongly nondegenerate self-concor-
dant on Ω ≔ ℝ>0, but it is not a barrier.
Solution. We already know that 𝑓 is self-concordant, since − log(𝑥) is self-concordant
(see Lecture 19), and addition of linear functions to self-concordant functions preserve
self-concordance.
The Hessian of 𝑓 is ∇2𝑓(𝑥) = 1/𝑥2, and the Newton step is correspondingly
𝑛(𝑥) = −[∇2𝑓(𝑥)]−1∇𝑓(𝑥) = −𝑥2(1 − 1
𝑥).
Hence, the intrinsic norm of the Newton step is
‖𝑛(𝑥)‖2
𝑥 = 1
𝑥2 [𝑥2(1 − 1
𝑥 )]
2
= 𝑥2 − 2𝑥 + 1,
which is unbounded as 𝑥 → +∞. □
L20.1.2 Complexity parameter and optimality gap of the central path
The complexity parameter of a barrier function is a crucial quantity that appears in the
analysis of interior-point methods. We now begin with its first application in providing an
upper bound on the optimality gap of the regularized problem.
Theorem L20.2. Let 𝑓 : Ω → ℝ be a barrier function. For any 𝛾 > 0, the point 𝜋(𝛾) on
the central path (see Definition L20.1), satisfies the inequality
⟨𝑐, 𝜋(𝛾)⟩ ≤ ( min
𝑥′∈Ω
⟨𝑐, 𝑥′⟩) + 1
𝛾 𝜃𝑓 .
The above result ensures that when 𝛾 becomes large enough, then the points on the central
path become arbitrarily close to the optimal value of the original problem. With little extra
work, the same can be said for approximate solutions to 𝜋(𝛾).
Theorem L20.3. Let 𝑓 : Ω → ℝ be a barrier function. For any 𝛾 > 0, and point 𝑥 ∈ Ω
such that ‖𝑥 − 𝜋(𝛾)‖𝑥 ≤ 1
6 ,
⟨𝑐, 𝑥⟩ ≤ ( min
𝑥′∈Ω
⟨𝑐, 𝑥′⟩) + 6
5𝛾 ⋅ 𝜃𝑓 .
L20.2 The (short-step) barrier method
The idea of the short-step barrier method is to chase the central path closely at every
iteration. This is conceptually the simplest interior point method, with more advanced
versions being the long-step barrier method and the predictor-corrector barrier method,
which is what is implemented in commercial solvers such as CPLEX and Gurobi. We will
use the term short-step barrier method and barrier method interchangeably today.
Assume that we know an initial point 𝑥1 ∈ Ω that is close to the point 𝜋(𝛾1) on the central
path, for some value of 𝛾1 > 0. The barrier algorithm now increases the parameter 𝛾1 to a
value 𝛾2 = 𝛽𝛾1 (where 𝛽 > 1), and applies Newton’s method to approximate the solution
𝜋(𝛾2). As long as 𝑥1 was sufficiently close to 𝜋(𝛾1), we expect that in switching from 𝛾1 to
𝛾2, the point 𝑥1 will still be in the region of quadratic convergence. In this case, Newton’s
method converges so fast, that (as we will see formally in the next subsection) a single
Newton step is sufficient to produce a point 𝑥2 ≔ 𝑥1 + 𝑛𝛾2 (𝑥1) that is again very close to
the central path at 𝜋(𝛾2). For the choice of parameter 𝛾2, the Newton step is in particular
𝑥2 ≔ 𝑥1 − [∇2𝑓(𝑥1)]−1(𝛾2𝑐 + ∇𝑓(𝑥1)),
since the objective function we apply the second-order descent direction is by definition the
problem
min
𝑥
s.t.
𝛾2⟨𝑐, 𝑥⟩ + 𝑓(𝑥)
𝑥 ∈ Ω.
Continuing this process indefinitely, that is,
𝛾𝑡+1 ≔ 𝛽𝛾𝑡, 𝑥𝑡+1 ≔ 𝑥𝑡 − [∇2𝑓(𝑥𝑡)]−1(𝛾𝑡+1𝑐 + ∇𝑓(𝑥𝑡))
we have the short-step barrier method.
L20.2.1 Update of the parameter 𝛾
As we did in Lecture 19, we will denote the second-order direction of descent—that is, the
Newton step—starting from a point 𝑥 using the letter 𝑛. However, since we are now dealing
with a continuum of objective functions parameterized on 𝛾, we will need to also specify
what objective (that is, what value of 𝛾) we are applying the Newton step to. For this
reason, we will introduce the notation
𝑛𝛾 (𝑥) ≔ −[∇2𝑓(𝑥)]−1(𝛾𝑐 + ∇𝑓(𝑥)).
The main technical hurdle in analyzing the short-step barrier method is to quantify the
proximity of the iterates to the central path. As is common with self-concordant functions,
we will measure such proximity using the lengths of the Newton steps: 𝑥𝑡 is near 𝜋(𝛾𝑡)
in the sense that the intrinsic norm of the Newton step 𝑛𝛾𝑡 (𝑥𝑡) is small (this should feel
natural recalling Theorem L19.6 in Lecture 19).
How close to the central path is close enough, so that the barrier method using a single
Newton update per iteration is guaranteed to work?
As we move our attention from the objective 𝛾𝑡⟨𝑐, 𝑥⟩ + 𝑓(𝑥) to the objective 𝛾𝑡+1⟨𝑐, 𝑥⟩ +
𝑓(𝑥), we can expect that distance to optimality of 𝑥𝑡 to 𝜋(𝛾𝑡+1) increases by a certain
amount compared to the distance from 𝑥𝑡 to 𝜋(𝛾𝑡). If this amount is not too large, then
we can hope to use Theorem L19.8 in Lecture 19 to “recover” in a single Newton step the
distance lost, and close the induction. The following theorem operationalizes the idea we
just stated, and provides a concrete quantitative answer to what “close enough” means. In
particular, we will show that ‖𝑛𝛾𝑡 (𝑥𝑡)‖𝑥𝑡 ≤ 1
9 is enough.
Theorem L20.4. If 𝑥𝑡 is close to the central path, in the sense that ‖𝑛𝛾𝑡 (𝑥𝑡)‖𝑥𝑡 ≤ 1
9 , then
by setting
𝛾𝑡+1 ≔ 𝛽𝛾𝑡 with 𝛽 ≔ (1 + 1
8√𝜃𝑓
),
the same proximity is guaranteed at time 𝑡 + 1, that is, ‖𝑛𝛾𝑡+1 (𝑥𝑡+1)‖𝑥𝑡+1 ≤ 1
9 .
Proof. We need to go from a statement pertaining ‖𝑛𝛾𝑡 (𝑥𝑡)‖𝑥𝑡 to one pertaining
‖𝑛𝛾𝑡+1 (𝑥𝑡+1)‖𝑥𝑡+1 . We will do so in two steps.
■ First part. Observe the equality (valid for all 𝛾𝑡+1 and 𝛾𝑡)
𝑛𝛾𝑡+1 (𝑥𝑡) = −[∇2𝑓(𝑥𝑡)]−1(𝛾𝑡+1𝑐 + ∇𝑓(𝑥𝑡))
= − 𝛾𝑡+1
𝛾𝑡
[∇2𝑓(𝑥𝑡)]−1(𝛾𝑡𝑐 + 𝛾𝑡
𝛾𝑡+1
∇𝑓(𝑥𝑡))
= − 𝛾𝑡+1
𝛾𝑡
[∇2𝑓(𝑥𝑡)]−1(𝛾𝑡𝑐 + ∇𝑓(𝑥𝑡)) + 𝛾𝑡+1 − 𝛾𝑡
𝛾𝑡
[∇2𝑓(𝑥𝑡)]−1∇𝑓(𝑥𝑡)
= 𝛾𝑡+1
𝛾𝑡
𝑛𝛾𝑡 (𝑥𝑡) + (𝛾𝑡+1
𝛾𝑡
− 1)[∇2𝑓(𝑥𝑡)]−1∇𝑓(𝑥𝑡).
Using the triangle inequality for norm ‖⋅‖𝑥𝑡 and plugging in the hypotheses of the
statement, we get
‖𝑛𝛾𝑡+1 (𝑥𝑡)‖𝑥𝑡 ≤ 𝛾𝑡+1
𝛾𝑡
‖𝑛𝛾𝑡 (𝑥𝑡)‖𝑥𝑡 + | 𝛾𝑡+1
𝛾𝑡
− 1| ⋅ ‖[∇2𝑓(𝑥𝑡)]−1∇𝑓(𝑥𝑡)‖𝑥𝑡
≤ 𝛾𝑡+1
𝛾𝑡
‖𝑛𝛾𝑡 (𝑥𝑡)‖𝑥𝑡 + | 𝛾𝑡+1
𝛾𝑡
− 1| ⋅ √𝜃𝑓
≤ 1
9(1 + 1
8√𝜃𝑓
) + 1
8√𝜃𝑓
√𝜃𝑓
≤ 1
9 ⋅ (1 + 1
8 ) + 1
8 = 1
4 (since 𝜃𝑓 ≥ 1).
However, the left-hand side of the inequality is ‖𝑛𝛾𝑡+1 (𝑥𝑡)‖𝑥𝑡 and not ‖𝑛𝛾𝑡+1 (𝑥𝑡+1)‖𝑥𝑡+1 .
This is where the second step comes in.
■ Second part. To complete the bound, we will convert from ‖𝑛𝛾𝑡+1 (𝑥𝑡)‖𝑥𝑡 to
‖𝑛𝛾𝑡+1 (𝑥𝑡+1)‖𝑥𝑡+1 . To do so, remember that 𝑥𝑡+1 is obtained from 𝑥𝑡 by taking a Newton
step. Hence, using Theorem L19.8 of Lecture 19, we have
‖𝑛𝛾𝑡+1 (𝑥𝑡+1)‖𝑥𝑡+1 ≤ ( ‖𝑛𝛾𝑡+1 (𝑥𝑡)‖𝑥𝑡
1 − ‖𝑛𝛾𝑡+1 (𝑥𝑡)‖𝑥𝑡
)
2
≤ (
1
4
1 − 1
4
)
2
= 1
9.
This completes the proof. □
Remark L20.1. Remarkably, a safe increase in 𝛾 depends only on the complexity
parameter 𝜃𝑓 of the barrier, and not on any property of the function. For example, for
a linear program
min
𝑥
s.t.
⟨𝑐, 𝑥⟩
𝐴𝑥 = 𝑏
𝑥 ∈ ℝ𝑛
≥0,
using the polyhedral barrier function, the increase in 𝛾 is independent of the number
of constraints of the problem or the sparsity of 𝐴, and we can increase 𝛾𝑡+1 = 𝛾𝑡 ⋅
(1 + 1
8√𝑛 ).
The result in Theorem L20.4 shows that at every iteration, it is safe to increase 𝛾 by a
factor of 1 + 1
8√𝜃𝑓
> 1, which leads to an exponential growth in the weight given to the
objective function of the problem.
Hence, combining the previous result with Theorem L20.2 we find the following guarantee.
Theorem L20.5. Consider running the short-step barrier method with a barrier func-
tion 𝑓 with complexity parameter 𝜃𝑓 , starting from a point 𝑥1 close to 𝜋(𝛾1), i.e.,
‖𝑛𝛾1 (𝑥1)‖𝑥1 ≤ 1/9, for some 𝛾1 > 0. For any 𝜀 > 0, after
𝑇 = ⌈10√𝜃𝑓 log( 6𝜃𝑓
5𝜀𝛾1
)⌉
iterations, the solution computed by the short-step barrier method guarantees an 𝜀-
suboptimal objective value ⟨𝑐, 𝑥𝑇 ⟩ ≤ ( min𝑥∈Ω⟨𝑐, 𝑥⟩) + 𝜀.
Proof. Since at every time the value of 𝛾 is increased by the quantity 1 + 1
8√𝜃𝑓
, the
number of iterations required to increase the value from 𝛾1 to any value 𝛾 is given by
𝑇 =
⌈
⌈
⌈
} log( 𝛾
𝛾1
)
log(1 + 1
8√𝜃𝑓
) ⌉
⌉
⌉

≤ ⌈log( 𝛾
𝛾1
) 5
4 ⋅ 8√𝜃𝑓 ⌉ (since 1
log(1 + 𝑥) ≤ 5
4𝑥 for all 0 ≤ 𝑥 ≤ 1
2)
= ⌈10√𝜃𝑓 log( 𝛾
𝛾1
)⌉.
On the other hand, we know from Theorem L20.3 that the optimality gap of 𝜋(𝛾) is given
by 6𝜃𝑓 /(5𝛾) as long as ‖𝑥𝑇 − 𝜋(𝛾𝑇 )‖𝑥𝑇
≤ 1
6 . This is indeed the case from Remark L19.3
of Lecture 19. So, to reach an optimality gap of 𝜀, we need 𝛾 = 6𝜃𝑓 /(5𝜀). Substituting
this value into the previous bound yields the statement. □
L20.2.2 Finding a good initial point
The result in Theorem L20.5 shows that, as long as we know a point 𝑥1 that is “close” (in
the formal sense of Theorem L20.4) to the central path, for a parameter 𝛾1 that is not too
small, then we can guarantee an 𝜀-suboptimal solution in roughly √𝜃𝑓 log(1/𝜀) iterations.
■ The analytic center. Intuitively, one might guess that a good initial point for the algorithm
would be a point close to 𝜁 ≔ 𝜋(0) (the minimizer of 𝑓 on Ω), which is often called the
analytic center of Ω. Let’s verify that that is indeed the case. By definition, such a point
satisfies ∇𝑓(𝜁) = 0, and so we have that
𝑛𝛾 (𝜁) = −𝛾[∇2𝑓(𝜁)]−1𝑐 ⟹ ‖𝑛𝛾 (𝜁)‖𝜁 = 𝛾 ⋅ ‖[∇2𝑓(𝜁)]−1𝑐‖𝜁 .
Hence, 𝑥1 = 𝜁 is within proximity 1/9 (in the sense of Theorem L20.4) of the central path
for the value of
𝛾1 = 1
9 ‖[∇2𝑓(𝜁)]−1𝑐‖𝜁
.
The only thing left to check is that 𝛾1 is not excessively small, so that the number of
iterations predicted in Theorem L20.5 is not too large. We now show that indeed we can
upper bound ‖[∇2𝑓(𝜁)]−1𝑐‖𝜁 .
Theorem L20.6. Let 𝜁 be the minimizer of the barrier 𝑓 on Ω. Then,
‖[∇2𝑓(𝜁)]−1𝑐‖𝜁 ≤ ⟨𝑐, 𝜁⟩ − min
𝑥∈Ω
⟨𝑐, 𝑥⟩.
(So, in particular, ‖[∇2𝑓(𝜁)]−1𝑐‖𝜁 ≤ max𝑥∈Ω ⟨𝑐, 𝑥⟩ − min𝑥∈Ω ⟨𝑐, 𝑥⟩.)
Proof. The direction −[∇2𝑓(𝜁)]−1𝑐 is a descent direction for 𝑐, since
⟨𝑐, −[∇2𝑓(𝜁)]−1𝑐⟩ = −‖[∇2𝑓(𝜁)]−1𝑐‖2
𝜁
≤ 0.
Hence, as we consider points 𝑥(𝜆) ≔ 𝜁 − 𝜆 ⋅ [∇2𝑓(𝜁)]−1𝑐 for 𝜆 ≥ 0 such that 𝑥(𝜆) ∈ Ω, we
have that the value of the objective ⟨𝑐, 𝑥(𝜆)⟩ decreases monotonically, and in particular
⟨𝑐, 𝑥(𝜆)⟩ = ⟨𝑐, 𝜁⟩ − 𝜆 ⋅ ‖[∇2𝑓(𝜁)]−1𝑐‖2
𝜁
,
which implies that
‖[∇2𝑓(𝜁)]−1𝑐‖2
𝜁
= ⟨𝑐, 𝜁⟩ − ⟨𝑐, 𝑥(𝜆)⟩
𝜆 ≤ ⟨𝑐, 𝜁⟩ − min𝑥∈Ω⟨𝑐, 𝑥⟩
𝜆 .
To complete the proof, it suffices to show that we can move in the direction of
−[∇2𝑓(𝜁)]−1𝑐 for a meaningful amount 𝜆. For this, we will use the property of self-
concordant function that the Dikin ellipsoid 𝑊 (𝜁) ≔ {𝑥 ∈ Ω : ‖𝑥 − 𝜁‖𝜁 < 1} ⊆ Ω. In
particular, this implies that any 𝜆 ≥ 0 such that
1 > ‖𝜁 − 𝑥(𝜆)‖𝜁 = 𝜆‖[∇2𝑓(𝜁)]−1𝑐‖𝜁
generates a point 𝑥(𝜆) ∈ Ω. So, we must have
‖[∇2𝑓(𝜁)]−1𝑐‖2
𝜁
≤ inf
{

⟨𝑐, 𝜁⟩ − min𝑥∈Ω⟨𝑐, 𝑥⟩
𝜆 : 0 < 𝜆 < 1
‖[∇2𝑓(𝜁)]−1𝑐‖𝜁 }

= (⟨𝑐, 𝜁⟩ − min
𝑥∈Ω
⟨𝑐, 𝑥⟩)‖[∇2𝑓(𝜁)]−1𝑐‖𝜁
,
which implies the statement. □
So, we have shown the following.
Theorem L20.7 (The analytic center 𝜁 is a good initial point). Let 𝑓 be a barrier function
with complexity parameter 𝜃𝑓 . If the short-step barrier method is initialized at the
analytic center 𝜁, then the number of iterations required to obtain an 𝜀-suboptimal
solution is bounded by
𝑇 = ⌈10√𝜃𝑓 log( 11 𝜃𝑓
𝜀 (⟨𝑐, 𝜁⟩ − min
𝑥∈Ω
⟨𝑐, 𝑥⟩))⌉.
■ Path switching and the auxiliary central path. In practice, we might not know where
the analytic center is. In this case, the typical solution is to first approximate the analytic
center, and then start the short step barrier method from there as usual.
To approximate the analytic center, one can use the auxiliary central path. The idea is the
following: start from an arbitrary point 𝑥′ ∈ Ω. Such a point is on the central path traced
by the solutions to
𝜋′(𝜈) ≔ arg min
𝑥
−𝜈⟨∇𝑓(𝑥′), 𝑥⟩ + 𝑓(𝑥)
s.t. 𝑥 ∈ Ω.
Indeed, note that 𝑥′ is the solution for 𝜈 = 1, that is, 𝑥′ = 𝜋′(1).
We can then run the short-step barrier method chasing 𝜋′ in reverse. At every step, we
will decrease the value of 𝜈 by a factor of 1 − 1
8√𝜃𝑓
. Once the value of 𝜈 is sufficiently small
that ‖[∇2𝑓(𝑥)]−1∇𝑓(𝑥)‖𝑥
≤ 1/6, we will have reached a point that is close to the analytic
center, and we can start the regular short-step barrier method for 𝜋(𝛾) from there. This
technique is called path switching, since we follow two central paths (one from 𝑥′ to the
analytic center, and one from the analytic center to the solution), switching around the
analytic center which path to follow. [▷ Try to work out the details and convince yourself
this works!]
L20.3 Further readings
The short book by Renegar, J. [Ren01] and the monograph by Nesterov, Y. [Nes18] (Chapter
5) provide a comprehensive introduction to self-concordant functions and their applications
in optimization.
I especially recommend the book by Renegar, J. [Ren01] for a concise yet rigorous account.
[Ren01] Renegar, J. (2001). A Mathematical View of Interior-point Methods in Convex
Optimization. SIAM. https://doi.org/10.1137/1.9780898718812
[Nes18] Nesterov, Y. (2018). Lectures on Convex Optimization. Springer International
Publishing. https://link.springer.com/book/10.1007/978-3-319-91578-4
Changelog
• Fixed a few typos (thanks Jonathan Huang!)
• Fixed a few typos (thanks Khizer!)
★These notes are class material that has not undergone formal peer review. The TAs and I are grateful
for any reports of typos.
¹Remember that lower-bounded self-concordant functions always have a unique minimizer, as seen in
Theorem L19.7 of Lecture 19.

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