Convex Optimization is one of the most important techniques in the field of mathematical programming, which has many applications. It also has much broader applicability beyond mathematics to disciplines like Machine learning, data science, economics, medicine, and engineering. In this blog post, you will learn about convex optimization concepts and different techniques with the help of examples.
What is convex optimization?
A convex optimization problem is an optimization problem where you want to find a point that maximizes/minimizes the objective function through iterative computations (typically, iterative linear programming) involving convex functions. The objective function is subjected to equality constraints and inequality constraints. Inequality constraints indicate that the solution should lie in some range whereas equality constraint requires it to lie exactly at a given point. Convex optimization can be used to also optimize an algorithm which will increase the speed at which the algorithm converges to the solution. It can also be used to solve linear systems of equations rather than compute an exact answer to the system. To solve convex optimization problems, machine learning techniques such as gradient descent are used.
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. Simpply 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 are of the following two types:
- Constrained convex optimization: The convex function to optimize is subject to convex constraints.
- Unconstrained convex optimization: The convex function to optimize is not subject to any convex constraints.
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.