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Does anyone know of such a library that performs mathematical optimization (linear programming, convex optimization, or more general types of problems)? I'm looking for something like MATLAB, but with the ability to handle larger problems. Do I have to write my own implementations, or buy one of those commercial products (CPLEX and the like)?
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You may try JOptimizer, open source and suitable for general convex optimization problems (linear programming, quadratic programming, qcqp, cone programming, semidefinite programming, ect
A good answer is dependent on what you mean by "convex" and "more general" If you are trying to solve large or challenging linear or convex-quadratic optimization problems (especially with a discrete component to them), then it's hard to beat the main commercial solvers, gurobi, cplex and Dash unless money is a big issue for you. They all have clean JNI interfaces and are available on most major platforms.
The coin-or project has several optimizers and have a project for JNI interface. It is totally free (EPL license), but will take more work to set-up and probably not give you the same performance.
OptaPlanner (Java, open source, ASL) can handle large problems and doesn't have an constraint type limitations (such as linear vs convex).
IPOPT has an interface for Java. You may also be able to adapt the APMonitor modeling language for Java. I develop this platform so I'll be glad to work with someone if they'd like to create a new interface to Java. It already has a Python API and MATLAB interface and includes solvers such as IPOPT, APOPT, BPOPT, and others that can handle large-scale systems.
Look into AMPL. The basic edition is free, but it costs money for larger problems. You don't pay for the language; you pay for solvers. It is also possible to upload your code and have it run on their servers.
There is a linear optimization tool called lpsolve. It's written in C (I think) but comes with a Java/JNI wrapper (API is not very OO but it does the job). It's pretty easy to use and I have had it running quite happily and stably in a live system for the last year.