how to solve convex optimization problems

• Convex Optimization

CSE 440: Introduction to Artificial Intelligence

Vishnu Boddeti

October 13, 2020

Content Credits: CMU AI, http://ai.berkeley.edu

Convex Optimization: Definition

• Convex Optimization Problem:
• A special class of optimization problem
• An optimization problem whose optimization objective $f$ is a convex function and feasible region $\mathcal{F}$ is a convex set.

Convex Combination

• A point between two points
• Given $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, a convex combination of them is any point of the form $\mathbf{z}=\theta\mathbf{x}+(1-\theta)\mathbf{y}$ where $\theta \in [0,1]$.
• When $\theta \in [0,1]$, $\mathbf{z}$ is called a strict convex combination of $\mathbf{x}$, $\mathbf{y}$.

Convex Sets

• Conceptually : Any convex combination of two points in the set is also in the set
• Mathematically : A set $\mathcal{F}$ is convex if $\forall x,y\in \mathcal{F}, \forall \theta \in [0,1]$,

Quiz: Convex Set

• Which of the following sets are convex?
• $\mathcal{F} = \mathbb{R}^n$
• $\mathcal{F} = \emptyset$
• $\mathcal{F} = \{\mathbf{x}_0\}, \mathbf{x}_0 \in \mathbb{R}^n$
• $\mathcal{F} = \mathcal{F}_1 \bigcap \mathcal{F}_2$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are convex sets.
• $\mathcal{F} = \mathcal{F}_1 \bigcup \mathcal{F}_2$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are convex sets.

Convex Function

• Value in the middle point is lower than average value
• Let $\mathcal{F}$ be a convex set. A function $f:\mathcal{F}\rightarrow\mathbb{R}$ is convex in $\mathcal{F}$ if $\forall x,y \in \mathcal{F}, \forall \theta \in [0,1]$,
• If $\mathcal{F}=\mathbb{R}^n$, we simply say $f$ is convex.

How to determine if a functions is convex?

• Prove by definition
• Use properties of convex functions
• Sum of convex functions is convex
• Convexity is preserved under a linear transformation
• If $f$ is a twice differentiable function of one variable, $f$ is convex on an interval $[a,b]\subseteq \mathbb{R}$ iff (if and only if) its second derivative $f''(x) \geq 0$ in $[a,b]$
• If $f$ is a twice continuously differentiable function of $n$ variables, $f$ is convex on $\mathcal{F}$ iff its Hessian matrix of second partial derivatives is positive semidefinite on the interior of $\mathcal{F}$.
• $H$ is positive semidefinite in $S$ if $\forall \mathbf{x} \in S, \forall \mathbf{z}\in\mathbb{R}^n, \mathbf{z}^T H(\mathbf{x}) \mathbf{z} \geq 0$
• $H$ is positive semidefinite in $\mathbb{R}^n$ iff all eigenvalues of $H$ are non-negative.
• Alternatively, prove $\mathbf{z}^T H(\mathbf{x})\mathbf{z} = \sum_i g_i^2(x,z)$

Concavity and Convexity

• Concave function
• A function $f$ is concave if $-f$ is convex
• Let $\mathcal{F}$ be a convex set. A function $f:\mathcal{F}\rightarrow\mathbb{R}$ is concave in $\mathcal{F}$ if $\forall \mathbf{x},\mathbf{y}\in\mathcal{F}, \forall \theta\in[0,1]$,
• The following is a convex optimization problem if $f$ is a concave function and $\mathcal{F}$ is a convex set

Quiz 2: Convex Function

• Which of the following functions are convex?
• $f(x) = e^{ax}, a\in\mathbb{R}$
• $f(x) = \log x, x\in(0,+\infty)$
• $f(x) = \|\mathbf{x}\|_2=\sqrt{\sum_{i}^nx_i^2}$
• $f(\mathbf{x},\mathbf{y})=\mathbf{x}^T\mathbf{A}\mathbf{y}$, $\mathbf{A}\in\mathbb{R}^{m\times n}$, $\mathbf{x}\in\mathbb{R}^{m}$, $\mathbf{y}\in\mathbb{R}^{n}$
• $f(x)=x^3, x\in\mathbb{R}$

Convex Optimization: Local Optima=Global Optima

• Given an optimization problem, a point $\mathbf{x}\in\mathbb{R}^n$ is globally optimal if $\mathbf{x}\in\mathcal{F}$ and $\forall \mathbf{y}\in\mathcal{F}$, $f(\mathbf{x})\leq f(\mathbf{y})$
• Given an optimization problem, a point $\mathbf{x}\in\mathbb{R}^n$ is locally optimal if $\mathbf{x}\in\mathcal{F}$ and $\exists R>0$ such that $\forall \mathbf{y}:\mathbf{y}\in\mathcal{F}$ and $\|\mathbf{x}-\mathbf{y}\|_2\leq R$, $f(\mathbf{x})\leq f(\mathbf{y})$
• Theorem 1 : For a convex optimization problem, all locally optimal points are globally optimal

Convex Optimization: How to Solve?

• Recall discrete setting
• Local search
• Iteratively improving an assignment
• Continuous and differentiable setting
• Iteratively improving value of $\mathbf{x}$
• Based on gradient
• For $f:\mathbb{R}^n\rightarrow\mathbb{R}$, gradient is the vector of partial derivatives
• A multi-variable generalization of the derivative
• Point in the direction of steepest increase in $f$

Gradient Descent

• Gradient descent: iteratively update the value of $\mathbf{x}$
• A simple algorithm for unconstrained optimization $\min_{\mathbf{x}\in\mathbb{R}^n} f(\mathbf{x})$
• How to choose $\mathbf{x}_0$, e.g., $\mathbf{x}_0=\mathbf{0}$
• How to choose and update step-size $\alpha$, e.g., trial and error, line-search methods etc.
• How to define "convergence", e.g., $\|\mathbf{x}^{i+1}-\mathbf{x}^i\| \leq \epsilon$

Projected Gradient Descent

• Iteratively update the value of $\mathbf{x}$ while ensuring $\mathbf{x}\in\mathcal{F}$
• $P_{\mathcal{F}}$ projects a point to the constraint set.
• How to choose $P_{\mathcal{F}}$, e.g., $P_{\mathcal{F}}=\underset{\mathbf{x'} \in \mathcal{F}}{\operatorname{arg min}}\|\mathbf{x}-\mathbf{x'}\|_2^2$
• Unconstrained and differentiable
• Gradient descent
• Set derivative to be 0
• Closed form solution
• (Not covered) Newton's Method (if twice differentiable)
• Constrained and differentiable
• (Not covered) Projected gradient descent
• (Not covered) Interior Point Method
• (Not covered) Non-differentiable
• $\epsilon$-Subgradient Method
• Cutting Plane Method

Convex Optimization: Apply

• Model a problem as a convex optimization problem
• Define variable, feasible set, objective function
• Prove it is convex (convex function + convex set)
• Solve the convex optimization problem
• Build up the model
• Call a solver
• Examples: fmincon (MATLAB), cvxpy (Python), cvxopt (Python), cvx (MATLAB)
• Map the solution back to the original problem

Convex Optimization

• Trial software
• Contact sales

What Is Convex Optimization?

Convex optimization is the process of minimizing a convex objective function subject to convex constraints or, equivalently, maximizing a concave objective function subject to convex constraints. Points satisfying local optimality conditions can be found efficiently for many convex optimization problems. Because a point that is a local optimum is also a global optimum, it is sufficient to find a local optimum to solve the problem. Convex approximations of nonconvex problems provide bounds on the optimal objective value and approximate solutions.

The figures below show examples of convex and nonconvex optimization problems.

Applications of convex optimization are found in finance and engineering, including portfolio optimization, design optimization, parameter estimation, signal processing, and optimal control. For example, the problem of selecting a portfolio of stocks to maximize return subject to upper bounds on risk and tracking error against a benchmark portfolio can be formulated as a convex optimization problem.

Convex optimization is the mathematical problem of finding a vector \(x\) that minimizes the function:

$$min_{x} f(x)$$

subject to:

\(g_i (x)≤0 \) (nonlinear inequality constraints)

\(Ax ≤b\) (linear inequality constraints)

\(A_{eq} x=b_{eq}\) (linear equality constraints)

\(lb ≤x ≤ub\) (bound constraints)

where \(g_i,i = 1,…,m\) are convex functions.

Linear programs (LP) and convex quadratic programs (QP) are convex optimization problems. Conic optimization problems, where the inequality constraints are convex cones, are also convex optimization problems. Problems with linear or convex quadratic objectives and linear and convex quadratic constraints (QCQP) can be represented as second-order cone programs (SOCP), which enables solving by efficient convex optimization methods.

Interior point algorithms are commonly used to solve convex optimization problems and can be written in MATLAB ® using matrix operations and the Cholesky factorization or the block LDL’ factorization . Optimization Toolbox™ has implementations of interior point algorithms for linear programs , quadratic programs , nonlinear programs , and second-order cone programs that are suitable for large-scale problems.

For more information on solving convex optimization problems, see Optimization Toolbox .

Examples and How To

• Constrained Electrostatic Nonlinear Optimization - Example
• Minimize Energy of Piecewise Linear Mass-Spring System - Example
• Optimization Models and Applications - Courseware
• Optimizers Everywhere—Optimization in Financial Applications with MATLAB - Slides

Software Reference

• Nonlinear Optimization - Documentation
• Quadratic Programming and Cone Programming - Documentation
• Linear Programming and Mixed-Integer Linear Programming - Documentation
• fmincon - Function
• coneprog - Function
• linprog - Function

See also: Optimization Toolbox , nonlinear programming , linear programming , quadratic programming , design optimization , portfolio optimization

9 MATLAB Cheat Sheets for Data Science and Machine Learning

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Optimization Problem Types - Convex Optimization

Optimization problem types, why convexity matters, convex optimization problems, convex functions, solving convex optimization problems.

• Other Problem Types

"...in fact, the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity."

- R. Tyrrell Rockafellar, in SIAM Review , 1993

Convex optimization problems are far more general than linear programming problems, but they share the desirable properties of LP problems:  They can be solved quickly and reliably up to very large size -- hundreds of thousands of variables and constraints. The issue has been that, unless your objective and constraints were linear, it was difficult to determine whether or not they were convex.  But Frontline System's Premium Solver Platform products includes an automated test for convexity of your problem functions.

A convex optimization problem is a problem where all of the constraints are convex functions , and the objective is a convex function if minimizing, or a concave function if maximizing.  Linear functions are convex , so linear programming problems are convex problems.  Conic optimization problems -- the natural extension of linear programming problems -- are also convex problems. In a convex optimization problem, the feasible region -- the intersection of convex constraint functions -- is a convex region, as pictured below.

With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal .  Several methods -- notably Interior Point methods -- will either find the globally optimal solution, or prove that there is no feasible solution to the problem.  Convex problems can be solved efficiently up to very large size.

A non-convex optimization problem is any problem where the objective or any of the constraints are non-convex, as pictured below.

Such a problem may have multiple feasible regions and multiple locally optimal points within each region.  It can take time exponential in the number of variables and constraints to determine that a non-convex problem is infeasible, that the objective function is unbounded, or that an optimal solution is the "global optimum" across all feasible regions.

Geometrically, a function is convex if a line segment drawn from any point (x, f(x)) to another point (y, f(y)) -- called the chord from x to y -- lies on or above the graph of f, as in the picture below:

Algebraically, f is convex if, for any x and y, and any t between 0 and 1, f( tx + (1-t)y ) <= t f(x) + (1-t) f(y).  A function is concave if -f is convex -- i.e. if the chord from x to y lies on or below the graph of f.  It is easy to see that every linear function -- whose graph is a straight line -- is both convex and concave.

A non-convex function "curves up and down" -- it is neither convex nor concave.  A familiar example is the sine function:

but note that this function is convex from -pi to 0, and concave from 0 to +pi.  If the bounds on the variables restrict the domain of the objective and constraints to a region where the functions are convex, then the overall problem is convex .

Because of their desirable properties, convex optimization problems can be solved with a variety of methods.  But Interior Point or Barrier methods are especially appropriate for convex problems, because they treat linear, quadratic, conic, and smooth nonlinear functions in essentially the same way -- they create and use a smooth convex nonlinear barrier function for the constraints, even for LP problems. 

These methods make it practical to solve convex problems up to very large size, and they are especially effective on second order (quadratic and SOCP) problems , where the Hessians of the problem functions are constant. Both theoretical results and practical experience show that Interior Point methods require a relatively small number of iterations (typically less than 50)  to reach an optimal solution, independent of the number of variables and constraints (though the computational effort per iteration rises with the number of variables and constraints).  Interior Point methods have also benefited, more than other methods, from hardware advances -- instruction caching, pipelining, and other changes in processor architecture.

Frontline Systems Solver Technology for Convex Problems

All Frontline Systems Solvers are effective on convex problems with the appropriate types of problem functions (linear, quadratic, conic, or nonlinear).  See Solver Technology for an overview of the available methods and Solver products. 

Convex Optimization Theory

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Weisstein, Eric W. "Convex Optimization Theory." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ConvexOptimizationTheory.html

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Convex optimization explained: Concepts & Examples

Prescriptive analytics plays a significant role in helping organizations make informed decisions by recommending the best course of action given a specific situation. Unlike descriptive and predictive analytics, which focus on understanding past data and predicting future outcomes, prescriptive analytics aims to optimize decision-making processes. Optimization solutions are a key component of prescriptive analytics , enabling decision-makers to find the most efficient and effective strategies to achieve their goals. 

Convex Optimization is a special class of optimization problems that deals with minimizing (or maximizing) convex functions over convex sets. Convex functions and sets exhibit specific mathematical properties that make them particularly well-suited for optimization. In the context of prescriptive analytics, convex optimization plays a key role in solving various optimization problems , such as resource allocation, scheduling, portfolio optimization , etc. It has much broader applicability beyond mathematics to disciplines like Machine learning , data science , economics, medicine, and engineering. By leveraging the unique properties of convex optimization, data scientists can develop efficient algorithms to find optimal solutions to real-world challenges and create robust prescriptive analytics solutions. In this blog post, you will learn about convex optimization concepts and different techniques with the help of examples.

Table of Contents

What is Convex Optimization?

Convex optimization is a powerful tool used to solve optimization problems in various fields such as finance, engineering, and machine learning. In a convex optimization problem, the goal is to find a point that maximizes or minimizes the objective function. This is achieved through iterative computations involving convex functions, which are functions that always lie above their chords.

The objective function is subject to both equality and inequality constraints . An equality constraint requires the solution to be exactly at a given point, while an inequality constraint restricts the solution to a certain range. These constraints are critical in defining the feasible region, which is the set of all solutions that satisfy the constraints.

Convex optimization can be used to optimize algorithms by improving the speed at which they converge to a solution. Additionally, it can be used to solve linear systems of equations by finding the best approximation to the system, rather than computing an exact answer.

Convex optimization plays a critical role in training machine learning models, which involves finding the optimal parameters that minimize a given loss function . In machine learning, convex optimization is used to solve a wide range of problems such as linear regression, logistic regression, support vector machines, and neural networks. The following are some applications of convex optimization in training machine learning models:

• Training linear regression models is a classic example of a convex optimization problem in which the goal is to find the best-fit line that minimizes the sum of squared errors between the predicted and actual values. The optimization problem can be formulated as a convex quadratic programming problem, which can be solved efficiently using convex optimization techniques.
• Training logistic regression models is another example of machine learning technique that involves finding the optimal parameters that maximize the likelihood of the observed data. This optimization problem can be formulated as a convex optimization problem with a log-concave objective function. The use of convex optimization techniques ensures that the optimization problem has a unique global minimum, making it more efficient to find the optimal solution.

Convexity plays an important role in convex optimizations . Convexity is defined as the continuity of a convex function’s first derivative. It ensures that convex optimization problems are smooth and have well-defined derivatives to enable the use of gradient descent. Some examples of convex functions are linear, quadratic, absolute value, logistic, exponential functions among others.

For convexity, convex sets are the most important. A convex set is a set that contains all points on or inside its boundary and contains all convex combinations of points in its interior. A convex set is defined as a set of all convex functions. Simply speaking, the convex function has a shape that is like a hill. A convex optimization problem is thus to find the global maximum or minimum of convex function. Convex sets are often used in convex optimization techniques because convex sets can be manipulated through certain types of operations to maximize or minimize a convex function. An example of a convex set is a convex hull, which is the smallest convex set that can contain a given convex set. A convex function takes the value only between its minimum and maximum values on any convex interval. This means that there exist no local extremes for this convex function (on the convex region). It also conveys that among all the points of this set which are on the convex hull, only one point is closest to the minimum.

Convex optimization problems can be broadly classified into the following two types:

• Constrained convex optimization : Constrained convex optimization involves finding the optimal solution to a convex function subject to convex constraints. These constraints may include both equality and inequality constraints. The objective function may be subject to a constraint that requires it to lie exactly at a given point or within a specified range. An example of a constrained convex optimization problem is portfolio optimization, where the goal is to find the optimal allocation of investments subject to constraints on risk and return.
• Unconstrained convex optimization : Unconstrained convex optimization involves finding the optimal solution to a convex function without any constraints. In this case, the objective function is simply a convex function that needs to be minimized or maximized. An example of an unconstrained convex optimization problem is linear regression, where the goal is to find the best-fit line that minimizes the sum of squared errors between the predicted and actual values.

What are different techniques that are used for convex optimization?

There are a variety of approaches for performing convex optimization. Some of them are following:

• Gradient methods using derivatives : One of the approaches to convex optimization is via first computing the convex function’s gradient, then using this derivative for performing convex optimization. If the convex problem has an easily computable convex objective function, then this gradient method of convex minimization can be used. This is because you are using information about the convex function to construct each step in the optimization. This convex optimization technique is widely used for solving convex problems.
• Projected gradient methods : This convex optimization approach was first introduced for solving convex problems with convex objective functions and convex constraints. It has an advantage over gradient methods using derivatives, if the system of convex constraints is not satisfied at the current iterate, it simply projects onto the subspace defined by these convex constraints.
• Quasi-Newton methods : This convex optimization approach is based on approximating the Hessian matrix of second derivatives by a quadratic approximation (where any convex function such as a linear one, can be used). The convex problem with convex constraints is solved using the quasi-Newton method to converge faster. It employs both first and second derivatives of convex objective functions to iteratively compute convex function minima using only function evaluations. The algorithm uses quadratic approximations around convex minimization iterations, thus requiring only function evaluations. An initial point is required to start convex optimization.
• Interior point methods using convex conjugate gradients and barrier function : If convexity is easy to check and convex function is twice differentiable, then the barrier function convex conjugate gradient method can be used to solve convex optimization problems. The two methods combined also make it easier to avoid going into a local minimum- there’s a gradient test before each iteration that checks if the step would go towards a local minimum.
• Global convex minimization : For convex problems, the quasi-Newton method usually gives better convergence results when compared with other techniques such as the gradient-based methods and the barrier function convex conjugate gradient method. However this might not always be the case, there might be convex problems where the quasi-Newton method converges slowly or even in a non-convergent manner. In such cases, an alternative approach is to use a global convex minimization technique instead of a local convex minimization technique.
• The method of multipliers : This convex optimization approach consists of iterations that change the original convex constraints into equality constraints by multiplying each constraint with a fixed multiplier so that the convex problem has convex objective functions and convex equality constraints at its minimization iterations.
• Modified optimization methods : The convex optimization problem can also be solved by modifications of Newton’s method. This convex optimization approach is suitable for convex functions which are only slightly convex, but not strongly convex. This convex optimization approach can be used to solve convex problems where the first or second derivative of convex objective functions cannot be computed easily.
• Reduced rank convex optimization : Convex functions can also be minimized by solving a reduced rank convex minimization problem. This is done by computing a convex function of a convex function. This convex optimization approach is suitable for convex functions with a bounded gradient, which means that the function has only finitely many critical points.
• Sequential convex programming : Sequential convex programming (SCP) is used for convex optimization problems that are convex but not strongly convex. It forms an iterative sequence that gradually improves the objective value, stopping when a user-specified termination criterion is satisfied or when no improvement to the objective function value is made within a user-specified number of iterations. The convex optimization problem is convex, even though it contains non-convex components.
• Likelihood ratio methods : For convex minimization problems, where convexity of the objective function is easy to check or convex constraints are convex but not strongly convex, likelihood ratio methods can be used to solve them. This convex optimization approach is suitable for convex functions which have a bounded gradient, convex constraints, and convex objective function with a single global minimizer.
• Stochastic optimization methods : In some cases, the convex optimization problem might not have an easily computable convex function with a Lipschitz continuous gradient. In those cases, gradients will not be used for convex optimization. Because of this, convex optimization problems can also be solved by simulation methods.
• Evolutionary algorithms : Evolutionary algorithms are not convex but they can still solve convex problems if the convex function has a bounded gradient or convex constraints with convex components or finite optimum points. This is done by perturbing the convex function components, which makes these convex optimization problems convex.

Real-world example of convex optimizations

Some real-life examples of convex optimization problems include the following:

• Scheduling of flights : Flight scheduling is an example convex optimization problem. It involves finding flight times that minimize costs like fuel, pilot/crew costs, etc. while maximizing the number of passengers.
• Facility location : Facility location problems are constructed to optimize the use of resources within a facility. These types of problems appear in many different fields such as logistics, telecommunications, and manufacturing.
• Inventory management : In inventory management, the optimization problem is to minimize the overall costs of ordering, holding, and shortage while maintaining stocks within the desired point.
• Routing phone calls : Convex programming has been applied to the general network design problem for routing telephone calls, which is an example convex optimization problem. The challenge with routing phone calls is finding the shortest path that connects a set of phone callers with a set of receivers.
• Fleet management : Convex optimization can be used to minimize the total operating costs of a fleet, subject to certain travel and scheduling restrictions.
• Logistics : Convex optimization problems in logistics deal with finding the number and type of vehicles that should be used for conveying goods from a set of agents (e.g. warehouses) to a set of customers (e.g. retail stores).
• Optimizing wind turbine output : Convex optimization has been applied to the control of wind turbines, where wind speed is modeled as a convex function of the turbine control variables.
• Energy consumption : Optimization of energy consumption in buildings is an example convex optimization problem that has been studied by many researchers. The convex objective function models the total cost to operate the building, which includes heating, cooling, lighting and other operational costs. Convex constraints are used to model restrictions on the control variables, such as occupancy sensors and timers.
• eCommerce : In eCommerce, convex optimization is used to minimize the cost of fulfilling orders. For example, minimizing the cost to fulfill an order while meeting demand constraints. It is also used in revenue management to maximize expected profit for a period of time given capacity and demand constraints.
• Campaign management : In marketing campaign management, convex optimization is used to determine the optimal locations and levels of customer service given fixed costs. It can also be used to find the optimal distribution channels for conveying information about products to customers, given fixed costs and expectations on consumer response.

In convex optimization, the function to be minimized or maximized is convex. A convex problem has an inequality constraint in which all variables are greater than or equal to zero (or alternatively less than or equal to zero). The gradient of this type of convex function will always point uphill, so it can’t get stuck at a local minimum that isn’t global. It’s also impossible for two gradients on opposite sides of the graph slope up and down in parallel because they’re not differentiable at their intersection point. This means that there is no need for iterations when using convex optimization techniques like machine learning algorithms – instead they work by simply moving downhill until reaching the optimum value. Convex problems can be solved on a convex minimization or convex maximization problem. Most machine learning algorithms like gradient descent, coordinate descent and batch gradient descent are used for convex optimization problems.

Ajitesh Kumar

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sir, Your article is very informative. Which optimization technique is best for frequent pattern mining of temporal data. Kindly suggest some techniques, as I am a Ph.D. scholar.

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Accelerated Optimization for Machine Learning: First-Order Algorithms

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