# Improving the Solving of Optimization Problems:
## 2 Solving Optimization Problems  
The goal of optimization problems is to determine the configuration of parameters, inputs, or variables that minimizes or maximizes an objective function. For example, optimization minimizes delivery times in logistics and reduces costs in energy systems. These problems arise in fields like engineering, finance, and healthcare, and solvers explore solutions in an intelligent and systematic manner.  
Optimization problems are classified based on variable types:  
* Combinatorial Optimization (CO): this involves discrete variables.
* Continuous Optimization: this involves continuous variables.  
When constraints limit feasible solutions, the problem is termed constrained optimization. Constraints can be integrated into the objective function via penalty functions or handled by generating only feasible solutions during exploration. The former is widely adopted in quantum optimization.  
Objective functions are called cost functions when minimized and fitness functions when maximized. Many real-world problems involve multi-objective optimization (MO), requiring trade-offs among objectives. Solutions can be identified using Pareto optimal sets, enabling preference evaluation during (interactive methods) or after (a posteriori methods) optimization. Alternatively, objectives can be aggregated into a single scalar function (a priori methods).  
Exploration strategies depend on problem characteristics. Brute-force approaches guarantee optimal solutions but become impractical as the problem size grows due to exponential complexity.  
Deterministic methods suit convex problems with single minima but face challenges in multimodal or non-convex scenarios. In contrast, heuristic approaches are preferred for large-scale problems, balancing the exploration of feasible spaces and the exploitation of prior solutions.  
Population-based algorithms explore solutions in parallel, inspired by natural and social systems. Examples include the following:* Cooperative algorithms: these mimic swarm behavior (e.g., particle swarm optimization [14]).
* Genetic algorithms (GAs): these model biological evolution [15].
* Artificial immune algorithms: these simulate immune responses [16].  
Algorithms inspired by physical systems are also prominent. For instance, simulated annealing (SA) [17] mimics annealing in materials science. SA probabilistically accepts solutions that may worsen objective values based on a temperature parameter, enabling exploration early and exploitation as temperature decreases. Modern hardware accelerators, such as probabilistic computers [18] and Coherent Ising Machines (CIMs) [19], exploit physical processes to solve optimization problems.  
Quantum solvers leverage quantum phenomena such as superposition, entanglement, and tunneling. Quantum annealing (QA) parallels simulated annealing (see Section 4), while quantum gate arrays (QGAs) use methods like Grover Adaptive Search and Quantum Approximate Optimization Algorithms (QAOAs) (see Section 5).  
However, quantum devices remain in the Noisy Intermediate-Scale Quantum (NISQ) era, with limited qubits, connectivity, and susceptibility to errors, posing challenges for real-world applications.