Questions tagged [nonconvex-programming]

For questions about non-convex optimization problems where the objective or any of the constraints are non-convex.

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2
votes
1answer
149 views

How to convexify log(convex) function?

I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $a$ and $b$ are ...
1
vote
1answer
47 views

Find an upper bound for an objective function

My objective function is $\log_2(1+{x^2y^2})$ and I found two upper bounds for $x^2$ and $y^2$. For example, assumed that we have the following upper bounds: $x^2\leq\text{constant}_1^2$ and $y^2\leq\...
1
vote
1answer
119 views

Non-linear optimization local or global solution

In an NLP, I have a constraint that I would like to formulate in a convex manner preferably without introducing binary variables and/or big M formulations if possible. The actual problem is non-convex ...
4
votes
0answers
61 views

How can non-polyhedral sets be investigated?

To derive problem-specific cutting planes for some given problem (think something like TSP problem), one common way is to study small examples. To this end, one can create small instances for the ...
4
votes
1answer
145 views

how to implement an optimization function with polynomial in Gurobi (Java)

I have the following problem: I have an objective function with the optimization variable $x$, which looks simplified like this: $ZF = (a+b)*(x+1)$ Here $a$ is simply a constant value. However, behind ...
3
votes
1answer
86 views

Maximizing a piecewise-linear convex function

Note: Initially posted on MathOverflow. I am working on an optimization problem where some of the terms of the objective function to maximize are expressed as a piecewise linear function of variables:...
10
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2answers
1k views

Trustful Nonlinear Programming

Is it possible for an NLP solver to claim that a knowingly feasible problem is infeasible? Shouldn't the solver be able to provide a solution (of course not necessarily the global optimum but a ...
3
votes
1answer
49 views

Maximize $\sum_{i=1}^n 1/x_i$ subject to an SDP constraint

I would like to solve the following problem: \begin{align}\max_{x_1, \ldots, x_n}&\quad\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}\\\text{s.t.}&\quad\sum_{i=1}^n x_i A_i \succeq A_0\...
3
votes
0answers
64 views

“Rank 1” type constraint $X=vw^\top$: MILP representation? Convex relaxation? Other tractable approach?

Suppose $X\in\mathbb{R}^{m\times n}$, $v\in\mathbb{R}^m$, $w\in\mathbb{R}^n$ are variables from an optimization problem, which also includes the constraints: $$0\le v\le a$$ $$0\le w\le 1$$ $$w_1+\...
3
votes
0answers
25 views

Stationary conditions for intersection

I wondered about this question for sometime. Definition of Stationarity (P) $\mbox{min} f(x)$ s.t $x\in C$ Let $f$ be $C^1$ function over a closed and convex set $C$ . then $x^*$ is called a ...
2
votes
3answers
333 views

Find the farthest point in hypercube to an exterior point

Let $\mathcal{U} = \{ [x_1, ..., x_n] \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$ be the unit hypercube and $C \in \mathbb{R}^n\setminus\mathcal{U}$ fixed. Let us consider the following problem $$ \max_{X ...
2
votes
1answer
76 views

Stationary condition for unit simplex

Consider the minimization problem $$\min_{x \in \Delta_n} f(x)$$ where $f$ is $C^1$ function over the unit simplex $\Delta_n$. Prove that $x^*\in\Delta_n$ is a stationary point of the problem iff ...
5
votes
0answers
69 views

Is there a way to use lazy constraints with Baron?

I am solving a non-linear mixed-integer programme with BARON. The objective function looks like $\big( \sum_i x_i \big) \cdot \big(\prod_i e^{-y_i}\big)$ (binary $x$ and real-valued $y$) and it has ...
2
votes
2answers
158 views

Pyomo + Ipopt. Speed Issue

I am using Pyomo + Ipopt as solver to solve a NLP problem. The problem is not extremely complex in terms of dimensionality and ...
3
votes
0answers
69 views

Appropriate Rotation Matrix in Nonconvex Optimization with Barrier

Let $ x \in \mathbb{R}^n_+$ be a variable such that $\sum_{i=1}^n x_i = 1$. In other words, $x$ is in a probability simplex. I am working on barrier-like functions in nonconvex optimization over such ...
6
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0answers
86 views

Cases where RLT/SDP relaxation does not work well with standard quadratic optimization

(For people who don't know what RLT is): I am maximizing an indefinite quadratic function over a standard simplex, i.e., the standard quadratic optimization problem. A well-known approach is to relax ...
4
votes
0answers
80 views

Fast solvers for LASSO-type non-convex optimization problems

Given $y \in \mathbb{R}^{n \times 1}, X \in \mathbb{R}^{n \times p}$, $p > n$, assume a LASSO-type optimization problem in the form of $$ \hat\beta=\underset{\beta}{\operatorname{argmin}}\frac{1}{2}...
6
votes
1answer
131 views

Convexity of the variance of a mixture distribution

$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words: $$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$ where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
7
votes
1answer
102 views

Minimizing sum of functions with pairwise dependence

I have formulated a problem where I need to minimize the sum of $N$ functions, with only pairwise dependence between the functions (any single constraint involves only two functions having adjacent ...
6
votes
1answer
91 views

Does strong duality hold when I dualize only a subset of the constraints?

Suppose I know that for some non-convex program: \begin{align}\min_x&\quad f(x)\\\text{s.t.}&\quad g_i(x)\leq 0, i \in C\end{align} strong duality holds for this problem. Now, suppose I form ...
8
votes
2answers
118 views

(Iterative?) Solutions to a certain quadratic program with non-convex constraints

Let $y\in\mathbb{R}^m$, $\tau\in\mathbb{R}$ and $X\in\mathbb{R}^{m\times n}$, with $\tau>0$ I would like to efficiently solve the following problem: Problem 1 Choose $\alpha,z\in\mathbb{R}^m,\beta\...
2
votes
1answer
74 views

AdaGrad - Sparsity of parameters

I read on Wikipedia: AdaGrad (for adaptive gradient algorithm) is a modified stochastic gradient descent algorithm with per-parameter learning rate, first published in 2011. Informally, this ...
6
votes
2answers
109 views

Local optimum of dual of non-linear program

In general, suppose you have a non-convex optimization problem with constraints and you form the dual problem. If you find a local optimum for the dual problem, will the corresponding primal solution ...
4
votes
1answer
56 views

Maximizing 1-norm: using binary variables to relax non-convexity

It is well-known that when we maximize a 1-norm, e.g., $\|Ax\|_1$, we can use binary variables and obtain a mixed-integer convex problem (otherwise maximizing 1-norm is non-convex). I am mentioning ...
3
votes
0answers
124 views

SDP relaxation with greater-than and less-than inequalities at the same time

I am dealing with the following nonconvex fractional quadratic optimization problem \begin{align} & \min_{\boldsymbol{x}} && \max_{t \in \mathcal{T}} \frac{\boldsymbol{a}_t^T \boldsymbol{...
3
votes
1answer
172 views

Supported pyomo free solvers for (non-convex) quadratic programming

Any one had the chance to use pyomo with free/open-source solvers that handle quadratic optimization problems, which they could be convex or not, but preferably as general as possible.
3
votes
1answer
169 views

Approximation methods for a mixed integer convex optimization problem

I have a convex objective function, e.g., minimizing the negative entropy function. My constraints are also linear. The only issue is that I also have binary variables. I am currently aware of AIMMS'...
4
votes
1answer
114 views

Minimize a convex function over a sphere

Problem description Let $\mathcal{C} = \{X \in \mathbb{R}^n \mid g(X) \leq 0\}$ with $g(X)$ a convex function. Suppose I need to solve the feasibility problem, for a given $r>0$ $$ \exists ^?X \...
3
votes
0answers
73 views

Strong Duality and Slater Condition

I am studying the Duality Chapter of Convex Optimization by Boyd. Is it possible that strong duality holds for non-convex optimization? If yes, is there any specific condition? And, what is the ...
4
votes
1answer
78 views

Dealing with a non-convex problem

I have the following objective function. The variables: $h_p$, $e_{trs}\left(h_p\right), w_{trs}\left(h_p\right)$ are all non-negative continuous. $T,R,S,\pi_{trs}$ are polynomially-sized sets. All ...
6
votes
1answer
135 views

Linearizing the square root of binary summations

My question is similar to this one and almost identical with this. I am so confused due to indexing and could not make sure if I could apply the solution in here to this indexed version as shown below....
6
votes
1answer
61 views

Linearizing the square root of two binary summations

My question is similar to this one though a bit more complicated. Though my question also includes indices, I am removing them to ease readability. Let binary variables $x,y\in\{0,1\}$, non-negative ...
13
votes
2answers
1k views

Is This Constraint Convex?

I have a constraint that I believe to be convex and not affine which I think means that I can implement a relaxation. I will first define the full constraint, and then build up my (informal) reasoning ...
11
votes
1answer
436 views

Solvers and saddle points

It seems like most solvers that can tackle nonlinear nonconvex optimization problems (e.g. IPOPT) operate on ultimately solving for the first-order optimality conditions. Can it therefore be assumed ...
9
votes
3answers
545 views

Examples of problems with non-convex constraint functions but convex feasible region

I'm looking for examples of (classes of) problems with a non-convex, non-linear formulation, but convex feasible region. That is, a problem of the sort: $$ \begin{array}{lll} \text{minimize} & ...
16
votes
1answer
764 views

IPOPT with HSL vs MUMPS

What are the advantages (if any) of using IPOPT with HSL vs MUMPS? HSL has a reputation of being faster, but does it walk the walk? In particular, does HSL scale better for large-scale problems? We ...
11
votes
0answers
138 views

Characterizing the solution of a (non) linear maximization program

I have the following maximization program \begin{align} \max\limits_{\{q_i\}}&\quad\sum\limits_{i=1}^nq_i \\ \text{s.t.}&\quad\begin{cases} k_j \geq \sum\limits_{i=1}^n q_i^{1 \over \...
9
votes
1answer
160 views

Rules of thumb for using classic methods (e.g. branch and bound) vs. meta-heuristics (G.A, S.A, etc..) for non-convex problems?

Are there any rules of thumb for when to use classical methods like branch and bound, branch and cut, etc...for non-convex problems, vs using meta-heuristic methods like genetic algorithms, ...
15
votes
1answer
298 views

Convexity of Variance Minimization

$X$ is a discrete random variable taking value $x_n$ with probability $1/N$ for $n=1, \ldots,N$. I would like to set the $x_n$ values in an optimization problem. My objective is to minimize the ...
12
votes
2answers
289 views

Convex vs Strictly Quasiconvex Functions in Optimization

I have read/heard quite a few time that in the old days, it was considered that linear programs constitute the class of optimization problems that can be solved efficiently in practice (as a rule of ...
14
votes
1answer
446 views

How to formulate a problem to prove/disprove convexity?

Given a general non-linear problem: \begin{align}P:\qquad&\min_{x\in X} f(x)\\\text{s.t.}\qquad&g(x)\leq 0\end{align} where $f$ is a non-linear function, $g$ is a vector of non-linear ...
10
votes
2answers
144 views

Global optimality condition of non-convex quadratic programs

We know that a convex quadratic maximization (not minimization!) on a polyhedron has its global optimal value on a vertex. Also, I have read in some papers that checking whether a vertex is globally ...