Optimization Theory Series: 7 — Convex Optimization and Non-convex Optimization

9 min readJan 10, 2024

In the previous article of our Optimization Theory series, titled “Optimization Theory Series: 6 — Linear and Quadratic Programming”, we delved into the fundamental concepts, solution methods, and various applications of linear and quadratic programming. These optimization problems, known for their special structural properties, offer intuitive and efficient solution methods. In this article, we will venture into a broader and more complex field of optimization theory: Convex Optimization and Non-convex Optimization.

Convex and non-convex optimizations play pivotal roles in modern mathematics and engineering. Not only are they significant branches of theoretical research, but they also form the cornerstone of numerous practical problem-solving scenarios. Convex optimization, renowned for its elegant mathematical properties and efficient solution strategies, finds widespread applications in areas such as machine learning, control theory, and financial engineering. Non-convex optimization, despite its greater challenges in both theory and practice, shows unparalleled capability in addressing complex real-world problems, especially evident in fields like big data analysis and neural network training.

Through this article, you will gain insights into the basic concepts and distinctions of convex and non-convex optimizations and their practical applications in solving real-world issues. Moreover, the article will discuss strategies for transitioning between these two types of optimizations, presenting a comprehensive perspective on optimization theory.

Stay tuned to our Optimization Theory series, as the next article will explore the world of “Integer Programming”, elucidating its concept and applications in combinatorial optimization and other areas.

At the end of this article, we will also briefly introduce some important yet unmentioned knowledge points related to the main topic, providing an avenue for readers with a keen interest in optimization theory to delve deeper.

Let’s embark on a journey to unravel the mysteries of convex and non-convex optimization!

Convex Optimization

Convex optimization is a significant branch of optimization theory, focusing on problems defined on convex sets with convex functions. To comprehend convex optimization, we first need to understand two fundamental concepts: Convex Sets and Convex Functions.

Defining Convex Sets and Convex Functions

Convex Sets: In Euclidean space, a set is called convex if the line segment between any two points within the set lies entirely within the set. Formally, a set C is convex if, for any x, y in C and any θ satisfying 0 ≤ θ ≤ 1, the point θx + (1 — θ)y is also in C.
Convex Functions: A function f defined on a convex set D is called a convex function if, for all x, y in D and 0 ≤ θ ≤ 1, it holds that f(θx + (1 — θ)y) ≤ θf(x) + (1 — θ)f(y). Intuitively, this means the chord line of the function always lies above or on the graph of the function.

Characteristics and Importance of Convex Optimization Problems

A key feature of convex optimization problems is that any local optimum is also a global optimum. This greatly simplifies the problem-solving process, as it means that local search techniques can be used to find a global optimum. Moreover, convex optimization problems generally possess well-behaved mathematical properties and efficient algorithms, making them critically important in both theoretical studies and practical applications.

Applications of Convex Optimization

Convex optimization has a wide range of applications across various fields. Some typical examples include:

Machine Learning and Data Science: In machine learning, many learning problems are formulated as convex optimization problems, like Support Vector Machines (SVM) and logistic regression, where convex optimization provides efficient and reliable solutions.
Control Theory: In the design of control systems, convex optimization is used to find optimal control strategies, especially within the framework of Linear Matrix Inequalities (LMI).
Signal Processing: Convex optimization is employed for efficient signal reconstruction in signal processing, evident in areas like compressed sensing and filter design.
Financial Engineering: In financial markets, convex optimization techniques are utilized for portfolio optimization and risk management, aiding investors to find an optimal balance between risk and return.

Through these applications, we observe that convex optimization is not only theoretically elegant but also immensely valuable in solving practical problems. It offers a powerful tool for scientists and engineers to tackle complex optimization challenges.

In the following section, we will explore the realm of non-convex optimization, a field that is more complex but equally rich in challenges and opportunities.

Non-convex Optimization

Non-convex optimization deals with optimization problems that do not satisfy the conditions of convex optimization. These problems are notorious for their complexity and the challenges they present in solving them. To understand non-convex optimization, let’s first clarify what non-convex sets and non-convex functions are.

Defining Non-convex Sets and Non-convex Functions

Non-convex Sets: A set is considered non-convex if there exists at least one pair of points in the set for which the line segment connecting them does not lie entirely within the set. Simply put, a non-convex set does not meet the criteria of a convex set.
Non-convex Functions: A function is non-convex if it is defined on a non-convex set or if it does not fulfill the definition of a convex function (i.e., there exist some x, y for which f(θx + (1 — θ)y) > θf(x) + (1 — θ)f(y)).

Challenges and Solutions in Non-convex Optimization Problems

The main challenge with non-convex optimization problems is that they do not guarantee that a local optimum is also a global optimum, making it exceedingly difficult to find the global best solution. Additionally, solving these problems can encounter multiple local optima, saddle points, or other complex local structures.

To address these challenges, various strategies have been developed, such as:

Heuristic methods: These include genetic algorithms, simulated annealing, etc., which use random searches to attempt to find solutions close to the global optimum.
Branch and bound methods: This systematic approach is used for iteratively narrowing down the search range for the global optimum.
Relaxation techniques: Transforming a non-convex problem into a convex approximation to simplify the problem.
Variants of gradient descent: Methods like momentum gradient descent, adaptive learning rate techniques, etc., are used to improve the performance of gradient descent in non-convex environments.

Applications of Non-convex Optimization in Real-world Problems

Despite the theoretical and computational complexities, non-convex optimization plays an indispensable role in practical applications. Some examples include:

Deep Learning: Training neural networks often involves non-convex optimization, as the error surface of a network is highly non-convex.
Optimization of Energy Systems: Such as designing and operating power grids, these problems often contain non-convex constraints and objective functions.
Engineering Design Problems: In fields like mechanical design and aerospace, optimization problems are often non-convex due to complex physical constraints and design goals.

These applications of non-convex optimization demonstrate its importance and effectiveness in tackling complex real-world problems. Though non-convex problems pose greater challenges, they also inspire new research and innovative solutions.

In the next section, we will compare convex and non-convex optimization, helping readers understand the differences and connections between these two optimization methods.

Comparing Convex and Non-convex Optimization

After exploring both convex and non-convex optimizations in the previous sections, it’s time to draw some comparisons between these two, particularly in terms of problem-solving difficulty and application domains.

Comparison in Problem-Solving Difficulty

Convex Optimization: Thanks to its mathematical properties, convex optimization problems are generally easier to solve. A key characteristic of convex optimization is that any local optimum is also a global optimum. This simplifies the solving process significantly, as algorithms only need to find a local optimum. Additionally, there are well-established solving algorithms for convex problems, such as interior-point methods and subgradient methods.
Non-convex Optimization: In contrast, non-convex optimization problems are typically more challenging to solve. This is because non-convex problems may have multiple local optima, and algorithms might get stuck in one of these local optima instead of finding the global optimum. Moreover, the solving process for non-convex problems can be more unstable and unpredictable.

Comparison in Application Domains

Convex Optimization: Primarily used in areas with clear convex properties, such as linear programming, certain types of machine learning problems, network flow issues, etc.
Non-convex Optimization: Suitable for more complex and irregular problems, like deep learning, nonlinear dynamical systems, certain engineering design challenges, etc. These complexities and non-convex natures require more advanced optimization techniques.

Strategies for Transitioning Between Convex and Non-convex Optimization

In some cases, non-convex problems can be transformed into convex problems, or at least approximated as convex problems. This transformation can simplify the solving process, although it may result in the loss of some original problem characteristics. Common strategies for such transformations include:

Relaxation Techniques: By relaxing some constraints, non-convex problems can be transformed into convex problems. For instance, in certain integer programming problems, relaxing the integer constraints can convert them into more solvable linear programming problems.
Approximation Methods: Finding a convex approximation for a non-convex problem, which may involve modifications to the objective function or constraints.
Piecewise Linearization: For some nonlinear problems, breaking down the nonlinear function into multiple linear segments to approximate, thereby transforming the problem into a convex optimization problem.

Overall, convex and non-convex optimizations differ significantly in terms of problem-solving difficulty and application domains. Understanding these differences is crucial for choosing the appropriate optimization strategy and algorithm.

In the next section, we will further illustrate these two types of optimization methods through some real-world case studies.

Real-world Case Studies

To better understand the application of convex and non-convex optimization in real-world scenarios, let’s examine a few case studies that illustrate their use in solving actual problems.

Case Study in Convex Optimization: Portfolio Optimization

Portfolio optimization in financial engineering is a classic example of convex optimization. The goal is to allocate investments among a set of assets to maximize returns and minimize risk. This problem can be formalized as a convex optimization problem. Investors aim to maximize expected returns while controlling risk, typically measured by variance.

In this context, both the objective function (expected return) and the constraints (risk control) can be constructed as convex functions. By applying convex optimization techniques, such as quadratic programming, investors can find the optimal asset allocation. This process is not only efficient but also reliable, as in convex optimization problems, any local optimum is also a global optimum.

Case Study in Non-convex Optimization: Deep Learning Model Training

Training deep learning models is a quintessential example of non-convex optimization. The objective is to adjust the weights and biases of a neural network to minimize prediction error. Due to the complexity of neural networks (e.g., nonlinear activation functions), the error surface is typically highly non-convex.

This means there are many local minima, and finding the global minimum is challenging. In practice, gradient-based methods (such as stochastic gradient descent and its variants) are often used to train the networks. Although these methods cannot guarantee finding a global optimum, they usually find sufficiently good solutions, allowing the trained models to perform well in practical applications.

Comparison and Discussion

These two case studies showcase the different facets of convex and non-convex optimization in practical applications. In portfolio optimization, convex optimization offers a stable and efficient solution. In contrast, while non-convex optimization faces more challenges, such as getting trapped in local optima, it can handle more complex and powerful models, like those in deep learning.

Through these real-world examples, we can see the strengths and suitable scenarios for both convex and non-convex optimization. Understanding these differences is essential for choosing the most appropriate optimization method to solve specific problems.

Conclusion

In this article, we have delved deeply into the concepts, characteristics, and applications of both convex and non-convex optimization, as well as their differences. Convex optimization is known for its simplicity in structure and efficiency in problem-solving, making it the preferred method for many domains. On the other hand, non-convex optimization, though more complex and challenging in solutions, proves highly effective in handling complex real-world problems.

The key advantage of convex optimization lies in the fact that any local optimum also serves as a global optimum, greatly reducing the complexity of the solution.
Non-convex optimization, while potentially presenting multiple local optima, demonstrates its strength in addressing complex problems like deep learning.

This discussion aims to provide readers with a better understanding of the application scenarios and significance of these two optimization methods.

In the next installment of our Optimization Theory series, we will explore “Integer Programming”. Integer programming is another critical branch of optimization theory, extensively applied in various practical scenarios, especially in combinatorial and discrete optimization. We will introduce the basic concepts, types, and practical applications of integer programming.

An important aspect not exhaustively discussed in this article is the application of “Duality Theory” in convex optimization. Duality theory is key to understanding the deeper structure of convex optimization, involving the transformation of the original optimization problem into what is known as the dual problem. This can help in gaining a deeper understanding of the nature of the original problem. In some cases, solving the dual problem can even be more efficient than addressing the original one.